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Equals an infinite convolution product_(1, 1, 1, ...)*(1, 0, 1, 0, 1, ...)*(1, 0, 0, 1, 0, 0, 1, ...)*(1, 0, 0, 0, 1, 0, 0, 0, 1, ...)*...; = a*b*c*...; where a = (1/(1-x)), b = (1/(1-x^2)), c = (1/(1-x^3)), etc. An array by rows: row 1 = a, row 2 = a*b, row 3 = a*b*c, ...; gives:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... = (a)
1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ... = (a*b)
1, 1, 2, 3, 4, 5, 7, 8, 10, 11, ... = (a*b*c)
1, 1, 2, 3, 4, 5, 6, 9, 11, 17, ... = (a*b*c*d)
1, 1, 2, 3, 5, 5, 7, 10, 13, 18, ... = (a*b*c*d*e)
1, 1, 2, 3, 5, 7, 11, 14, 20, 25, ... = (a*b*c*d*e*f)
1, 1, 2, 3, 5, 7, 11, 15, 21, 27, ... = (a*b*c*d*e*f*g)
1, 1, 2, 3, 5, 7, 11, 15, 22, 28, ... = (a*b*c*d*e*f*g*h)
1, 1, 2, 3, 5, 7, 11, 15, 22, 29, ... = (a*b*c*d*e*f*g*h*i)
... with rows tending to A000041. Partition triangles A058398 = ascending antidiagonals. Partition triangle A008284 reversal of A058398. (End)
Starting with offset 1 = row sums of triangle A168532. - Gary W. Adamson, Nov 28 2009
P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...),
and A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...),
and A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. - Gary W. Adamson, Feb 11 2010
Equals row sums of triangle A173304. - Gary W. Adamson, Feb 15 2010
p(x) = A(x)*A(x^2), A(x) = A174065; p(x) = B(x)*B(x^3), B(x) = A174068. Equals row sums of triangles A174066 and A174067. - Gary W. Adamson, Mar 06 2010
Triangle A113685 is equivalent to p(x) = p(x^2) * A000009(x). Triangle A176202 is equivalent to p(x) = p(x^3) * A000726(x). - Gary W. Adamson, Apr 11 2010
A sequence of positive integers p = p_1 ... p_k is a descending partition of the positive integer n if p_1 + ... + p_k = n and p_1 >= ... >= p_k. If formally needed p_j = 0 is appended to p for j > k. Let P_n denote the set of these partition for some n >= 1. Then a(n) = 1 + Sum_{p in P_n} floor((p_1-1)/(p_2+1)). (Cf. A000065, where the formula reduces to the sum.) Proof in Kelleher and O'Sullivan (2009). For example a(6) = 1 + 0 + 0 + 0 + 0 + 1 + 0 + 0 + 1 + 1 + 2 + 5 = 11. - Peter Luschny, Oct 24 2010
Let n = Sum( k_(p_m) p_m ) = k_1 + 2k_2 + 5k_5 + 7k_7 + ..., where p_m is the m-th generalized pentagonal number (A001318). Then a(n) is the sum over all such pentagonal partitions of n of (-1)^(k_5+k_7 + k_22 + ...) ( k_1 + k_2 + k_5 + ...)! /( k_1! k_2! k_5! ...), where the exponent of (-1) is the sum of all the k's corresponding to even-indexed GPN's. - Jerome Malenfant, Feb 14 2011
From Jerome Malenfant, Feb 14 2011: (Start)
The matrix of a(n) values
a(0)
a(1) a(0)
a(2) a(1) a(0)
a(3) a(2) a(1) a(0)
....
a(n) a(n-1) a(n-2) ... a(0)
is the inverse of the matrix
1
-1 1
-1 -1 1
0 -1 -1 1
....
-d_n -d_(n-1) -d_(n-2) ... -d_1 1
where d_q = (-1)^(m+1) if q = m(3m-1)/2 = the m-th generalized pentagonal number (A001318), = 0 otherwise. (End)
Equals row sums of triangle A187566. - Gary W. Adamson, Mar 21 2011
Let k > 0 be an integer, and let i_1, i_2, ..., i_k be distinct integers such that 1 <= i_1 < i_2 < ... < i_k. Then, equivalently, a(n) equals the number of partitions of N = n + i_1 + i_2 + ... + i_k in which each i_j (1 <= j <= k) appears as a part at least once. To see this, note that the partitions of N of this class must be in 1-to-1 correspondence with the partitions of n, since N - i_1 - i_2 - ... - i_k = n. - L. Edson Jeffery, Apr 16 2011
a(n) is the number of distinct degree sequences over all free trees having n + 2 nodes. Take a partition of the integer n, add 1 to each part and append as many 1's as needed so that the total is 2n + 2. Now we have a degree sequence of a tree with n + 2 nodes. Example: The partition 3 + 2 + 1 = 6 corresponds to the degree sequence {4, 3, 2, 1, 1, 1, 1, 1} of a tree with 8 vertices. - Geoffrey Critzer, Apr 16 2011
a(n) is number of distinct characteristic polynomials among n! of permutations matrices size n X n. - Artur Jasinski, Oct 24 2011
Conjecture: starting with offset 1 represents the numbers of ordered compositions of n using the signed (++--++...) terms of A001318 starting (1, 2, -5, -7, 12, 15, ...). - Gary W. Adamson, Apr 04 2013 (this is true by the pentagonal number theorem, Joerg Arndt, Apr 08 2013)
a(n) is also number of terms in expansion of the n-th derivative of log(f(x)). In Mathematica notation: Table[Length[Together[f[x]^n * D[Log[f[x]], {x, n}]]], {n, 1, 20}]. - Vaclav Kotesovec, Jun 21 2013
Conjecture: No a(n) has the form x^m with m > 1 and x > 1. - Zhi-Wei Sun, Dec 02 2013
Partitions of n that contain a part p are the partitions of n - p. Thus, number of partitions of m*n - r that include k*n as a part is A000041(h*n-r), where h = m - k >= 0, n >= 2, 0 <= r < n; see A111295 as an example. - Clark Kimberling, Mar 03 2014
a(n) is the number of compositions of n into positive parts avoiding the pattern [1, 2]. - Bob Selcoe, Jul 08 2014 |
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0,1,3,4,7，…)——Gary W. Adamson, 2008年11月15日
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