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A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009). 19
1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Invert transform of A000009.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2816

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Partition Function Q

Index entries for sequences related to partitions

Index entries for sequences related to compositions

FORMULA

G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).

G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).

G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).

G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).

a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - Vaclav Kotesovec, May 23 2018

MAPLE

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(

     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

    end:

a:= proc(n) option remember; `if`(n=0, 1,

      add(b(j)*a(n-j), j=1..n))

    end:

seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

MATHEMATICA

nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]

a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

CROSSREFS

Cf. A000009, A050342, A055887, A067687, A081362, A089259, A270995, A279785, A299106.

Row sums of A308680.

Sequence in context: A006054 A106805 A094981 * A239812 A097779 A319768

Adjacent sequences:  A304966 A304967 A304968 * A304970 A304971 A304972

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 22 2018

STATUS

approved

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Last modified June 21 03:24 EDT 2021. Contains 345354 sequences. (Running on oeis4.)