A086209 -id:A086209 - cp's OEIS frontend

A007279 Number of partitions of n into partition numbers.

1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 44, 54, 66, 79, 95, 113, 133, 157, 184, 216, 250, 290, 335, 385, 442, 505, 576, 656, 743, 842, 951, 1070, 1204, 1351, 1514, 1691, 1887, 2102, 2336, 2595, 2875, 3184, 3519, 3883, 4282, 4713, 5181, 5690, 6241, 6839, 7482

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.

Cf. A000041.

Cf. A086209, A229362.

with(combinat): gf := 1/product((1-q^numbpart(k)), k=1..20): s := series(gf, q, 200): for i from 0 to 199 do printf(`%d,`,coeff(s, q, i)) od: # James Sellers, Feb 08 2002

CoefficientList[ Series[1/Product[1 - x^PartitionsP[i], {i, 1, 15}], {x, 0, 50}], x]

seq(n)={my(t=1); while(numbpart(t+1)<=n, t++); Vec(1/prod(k=1, t, 1-x^numbpart(k) + O(x*x^n)))} \\ Andrew Howroyd, Jun 22 2018

G.f.: 1/Product_{k>=1} (1-q^A000041(k)). - Michel Marcus, Jun 20 2018

More terms from James Sellers, Feb 08 2002

a(0)=1 prepended by Alois P. Heinz, Jul 02 2017

A280253 Expansion of Product_{k>=1} (1 + x^p(k)), where p(k) is the number of partitions of k (A000041).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 4, 4, 5, 5, 6, 5, 6, 6, 6, 6, 6, 7, 6, 7, 8, 7, 7, 8, 8, 9, 8, 9, 9, 9, 10, 9, 11, 9, 10, 12, 10, 11, 11, 11, 12, 11, 12, 13, 13, 14, 14, 14, 13, 13, 15, 15, 14, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 15, 17, 19, 19, 20, 19, 20, 20, 19, 21, 20, 20, 21, 21, 22

Offset: 0

Ilya Gutkovskiy, Dec 30 2016

nonn

Number of partitions of n into distinct partition numbers.

			a(8) = 3 because we have [7, 1], [5, 3] and [5, 2, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q
Index entries for related partition-counting sequences

Cf. A000009, A000041, A007279, A086209.

nmax = 90; CoefficientList[Series[Product[(1 + x^PartitionsP[k]), {k, 1, nmax}], {x, 0, nmax}], x]

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

cp's OEIS Frontend

A007279 Number of partitions of n into partition numbers.

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