A356269
a(n) = Sum_{k=0..n} binomial(2*k, k) * p(k), where p(k) is the partition function A000041.
Original entry on oeis.org
1, 3, 15, 75, 425, 2189, 12353, 63833, 346973, 1805573, 9565325, 49069517, 257289529, 1307750129, 6723491129, 34024174649, 172873744739, 865954792079, 4359881882579, 21679061144579, 108108834714719, 534409071271199, 2642716232918639, 12975671796056639, 63765647596939139
Offset: 0
-
Table[Sum[Binomial[2*k, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]
A324236
Expansion of e.g.f. 1 / (1 - Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).
Original entry on oeis.org
1, 1, 4, 21, 149, 1317, 13985, 173209, 2451844, 39044784, 690862770, 13446615722, 285510978887, 6567419023617, 162686428939423, 4317885767971448, 122241788335870103, 3677030054440996775, 117111150680951037907, 3937135961534144480556, 139328182441566999124409
Offset: 0
-
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-k)*
binomial(n, k)*combinat[numbpart](k), k=1..n))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 02 2019
-
nmax = 20; CoefficientList[Series[1/(1 - Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] PartitionsP[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
A294350
Product of first n terms of the binomial transform of the partition function (A000041).
Original entry on oeis.org
1, 2, 10, 130, 4420, 388960, 87516000, 49796604000, 70960160700000, 251057048556600000, 2188464292267882200000, 46682131818366195208200000, 2421822316605019841206207800000, 303875733353698259555507717497200000, 91748896295748761809334889636212098800000
Offset: 0
-
Table[Product[Sum[Binomial[m, k]*PartitionsP[k], {k, 0, m}], {m, 0, n}], {n, 0, 15}]
-
a(n) = prod(m=0, n, sum(k=0, m, binomial(m,k)*numbpart(k))); \\ Michel Marcus, Oct 29 2017
A307264
Expansion of (1/(1 - x)) * Product_{k>=1} 1/(1 + (-x)^k/(1 - x)^k).
Original entry on oeis.org
1, 2, 3, 5, 10, 22, 49, 107, 229, 486, 1035, 2225, 4825, 10508, 22875, 49624, 107154, 230356, 493471, 1054602, 2250850, 4801825, 10244940, 21865466, 46680201, 99659713, 212697816, 453634533, 966551216, 2057052465, 4372660927, 9284272791, 19692591418
Offset: 0
-
a:=series((1/(1-x))*mul(1/(1+(-x)^k/(1-x)^k),k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 03 2019
-
nmax = 32; CoefficientList[Series[1/(1 - x) Product[1/(1 + (-x)^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
Comments