A032310 -id:A032310 - cp's OEIS frontend

A334370 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k) / prime(k)!).

1, 0, 1, 1, 0, 11, 0, 22, 56, 36, 2640, 1, 8712, 79, 72436, 360465, 48608, 49008961, 794376, 4232764, 7753140, 942565890, 18198334, 14799637777, 10577976, 366619314900, 2785137222400, 1475339135400, 1065920156634060, 3765722000041, 5869315258699050

Offset: 0

Ilya Gutkovskiy, May 11 2020

nonn

a(n) is the number of functions f:[n]-> [n] such that the number of elements that are mapped to i is either 0 or the i-th prime. a(5) = 11: (33333), (11222), (12122), (12212), (12221), (21122), (21212), (21221), (22112), (22121), (22211). - Alois P. Heinz, Jul 18 2023

Alois P. Heinz, Table of n, a(n) for n = 0..836

Cf. A000040, A000720, A007837, A032310, A115278, A178682, A190476, A319113, A364344.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      (p-> `if`(p>n, 0, b(n-p, i-1)*binomial(n, p)))(ithprime(i))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 18 2023

nmax = 30; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[DivisorSum[k, -#/(-#!)^(k/#) &, PrimeQ[#] &] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 30}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+isprime(k)*x^k/k!))) \\ Seiichi Manyama, Feb 27 2022

A115278 Number of partitions of {1,...,2*n} into even sized blocks such that no block size is repeated.

Original entry on oeis.org

1, 1, 1, 16, 29, 256, 14422, 49141, 490429, 10758400, 1797335306, 9458619391, 133756636598, 2528529510391, 137864810180749, 53441183229799381, 410251032050409469, 7615997734377068128, 167055180095977694194, 6741819165851219788075, 738863335901972011745434

Offset: 0

Christian G. Bower, Jan 18 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..370

Cf. A005046, A007837, A032310, A115277.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!* b[n - i*j, i - 2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[2 n, 2 n]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

E.g.f.: B(x) of b(n) where b(2*n)=a(n), b(2*n+1)=0. B(x)=Product {m >= 1} (1+x^(2*m)/(2*m)!).

A294531 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)!).

Original entry on oeis.org

1, 1, 2, 7, 28, 141, 866, 6063, 48560, 438721, 4387582, 48272643, 579642328, 7535456657, 105499762356, 1582665820557, 25322712724800, 430488412249937, 7748929638924950, 147229720951176075, 2944597048114831688, 61836721841638907121, 1360407969674984670156

Offset: 0

Seiichi Manyama, Nov 02 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..449

Cf. A005651, A032310.

N:= 40:
S:= series(1/mul(1-x^j/j!,j=1..N,2),x,N+1):
seq(coeff(S,x,n)*n!,n=0..N); # Robert Israel, Nov 23 2017

nmax = 30; CoefficientList[Series[1/Product[(1-x^(2*k-1)/(2*k-1)!), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 02 2017 *)

A371550 Expansion of e.g.f. Product_{k>=1} (1 + mu(k)^2*x^k/k!).

Original entry on oeis.org

1, 1, 1, 4, 4, 11, 67, 29, 260, 876, 3841, 34134, 69774, 152231, 774243, 4182754, 30376720, 409813561, 1056300594, 3175397668, 3655126844, 91668397027, 499871922705, 5219638765816, 120716816247428, 17518596045460, 193032439198301, 3860666483055372, 22675538336311998

Offset: 0

Ilya Gutkovskiy, Mar 27 2024

nonn

Cf. A005117, A007837, A008683, A032310, A087188, A115278, A334370, A371549.

nmax = 28; CoefficientList[Series[Product[(1 + MoebiusMu[k]^2 x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A334385 E.g.f.: Product_{k>=1} (1 + x^(k(k + 1)/2) / (k(k + 1)/2)!).

Original entry on oeis.org

1, 1, 0, 1, 4, 0, 1, 7, 0, 84, 841, 11, 0, 286, 4004, 1, 8024, 136136, 816, 7775256, 155195040, 54265, 1193830, 0, 109832360, 2749077760, 84987760, 296010, 10716746041, 310545275069, 1201800600, 2444026056820, 77016647623040, 0, 14402113079955304, 504073957798435640

Offset: 0

Ilya Gutkovskiy, May 11 2020

nonn

Cf. A007837, A032310, A053614 (positions of 0's), A115278, A205799.

nmax = 35; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2)/(k (k + 1)/2)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[DivisorSum[k, -#/(-#!)^(k/#) &, IntegerQ[Sqrt[8 # + 1]] &] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 35}]

cp's OEIS Frontend

A334370 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k) / prime(k)!).

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A115278 Number of partitions of {1,...,2*n} into even sized blocks such that no block size is repeated.

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A294531 E.g.f.: 1/Product_{k>0} (1-x^(2*k-1)/(2*k-1)!).

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Maple

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A371550 Expansion of e.g.f. Product_{k>=1} (1 + mu(k)^2*x^k/k!).

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A334385 E.g.f.: Product_{k>=1} (1 + x^(k*(k + 1)/2) / (k*(k + 1)/2)!).

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A294531 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)!).

A334385 E.g.f.: Product_{k>=1} (1 + x^(k(k + 1)/2) / (k(k + 1)/2)!).