A007279 -id:A007279 - cp's OEIS frontend

A320844 Expansion of Product_{k>0} (1-x^p(k)), where p(k) is the number of partitions of k (A000041).

1, -1, -1, 0, 1, 0, 0, 0, 1, 0, -1, -1, 1, 1, -1, -2, 2, 2, -1, -2, 0, 1, -1, 0, 1, 2, 0, -2, -2, 2, -1, 0, 1, 2, -1, -1, 0, 2, -3, -2, 1, 3, -1, 0, 1, 3, -3, -4, 0, 4, 1, -3, 1, 2, -1, -4, -1, 5, 2, -4, 0, 3, 1, -3, -1, 0, 1, -3, 1, 3, 3, -2, -2, -2, 1, -1, 1, 1, 3, -3

Offset: 0

Seiichi Manyama, Oct 22 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A007279.

Cf. A000041, A280253.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1-x^NumberOfPartitions(k): k in [1..100]]))); // G. C. Greubel, Oct 27 2018

CoefficientList[Series[Product[1 - x^PartitionsP[k], {k, 1, 120}], {x, 0, 100}], x] (* G. C. Greubel, Oct 27 2018 *)

x='x+O('x^50); Vec(prod(k=1,50, 1-x^numbpart(k))) \\ G. C. Greubel, Oct 27 2018

A280254 Expansion of 1/(1 - Sum_{k>=1} x^p(k)), where p(k) is the number of partitions of k (A000041).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 50, 95, 180, 343, 652, 1240, 2358, 4484, 8528, 16217, 30840, 58649, 111532, 212101, 403352, 767056, 1458711, 2774031, 5275379, 10032192, 19078230, 36281088, 68995780, 131209344, 249520934, 474514204, 902384123, 1716064761, 3263442024, 6206090863, 11802129022, 22444120219

Offset: 0

Ilya Gutkovskiy, Dec 30 2016

nonn

Number of compositions (ordered partitions) into partition numbers.

			a(4) = 7 because we have [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Partition Function P
Index entries for sequences related to compositions

Cf. A000041, A007279.

nmax = 38; CoefficientList[Series[1/(1 - Sum[x^PartitionsP[k], {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^p(k)).

cp's OEIS Frontend

A320844 Expansion of Product_{k>0} (1-x^p(k)), where p(k) is the number of partitions of k (A000041).

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