A280253 -id:A280253 - cp's OEIS frontend

A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

1, 2, 3, 5, 6, 8, 11, 13, 16, 19, 22, 26, 30, 34, 38, 44, 49, 54, 62, 67, 74, 83, 89, 98, 107, 115, 124, 134, 145, 155, 168, 178, 189, 206, 217, 231, 247, 259, 277, 294, 310, 327, 345, 365, 382, 404, 424, 444, 470, 489, 513, 539, 561, 588, 613, 641, 670, 699, 729, 756, 791, 824

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Number of partitions of n into distinct terms of A000009, where 2 different parts of 1 and 2 different parts of 2 are available (1a, 1b, 2a, 2b, 3a, 4a, 5a, 6a, ...).

			a(3) = 5 because we have [3a], [2a, 1a], [2a, 1b], [2b, 1a] and [2b, 1b].

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for related partition-counting sequences

Cf. A000009, A007279, A050342, A068006, A089254, A089259, A280253, A284908.

N:= 20: # to get a(0) .. a(A000009(N))
P:= mul(1+x^k,k=1..N):
R:= mul(1+x^coeff(P,x,n)),n=1..N):
seq(coeff(R,x,n),n=0..coeff(P,x,N)); # Robert Israel, Sep 01 2017

nmax = 61; CoefficientList[Series[Product[1 + x^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000009(k)).

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

Original entry on oeis.org

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

A360093 a(n) is the smallest positive integer which can be represented as the sum of distinct partition numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 3, 8, 15, 18, 23, 30, 33, 38, 43, 45, 48, 56, 58, 63, 71, 74, -1, 78, 80, 85, 90, 93, 100, 101, 106, 104, 109, 113, 115, 119, 122, 130, -1, 134, 135, 145, 141, 150, 153, 146, 149, 163, 156, 158, 165, 167, 173, -1, 176, 178, 182, 181, -1, 183, 186, 196, 193, 191, 199

Offset: 1

Ilya Gutkovskiy, Jan 25 2023

sign

			a(5) = 18 because we have [15, 3], [15, 2, 1], [11, 7], [11, 5, 2] and [7, 5, 3, 2, 1].

Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q

Cf. A000041, A280253.

cp's OEIS Frontend

A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

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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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A360093 a(n) is the smallest positive integer which can be represented as the sum of distinct partition numbers in exactly n ways, or -1 if no such integer exists.

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