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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 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.)