A294373 -id:A294373 - cp's OEIS frontend

A058807 a(n) = Product_{k=1..n} s(n,k), where s(n,k) is unsigned Stirling number of the first kind. (s(n,k) = number of permutations of n elements which contain exactly k cycles.)

1, 1, 6, 396, 420000, 9432450000, 5571367220160000, 103458225408290423193600, 70288262635020872178876253470720, 1993179010286886206697449779415040000000000, 2650683735711909138223088071500675703191552000000000000

Offset: 1

Leroy Quet, Jan 02 2001

			a(4) = s(4,1)*s(4,2)*s(4,3)*s(4,4) = 6*11*6*1 = 396.

Vaclav Kotesovec, Table of n, a(n) for n = 1..36

Cf. A058808, A132393, A294373.

a:=n->mul(abs(Stirling1(n, k)), k=1..n): seq(a(n), n=1..10); # Zerinvary Lajos, Jun 28 2007

Abs[Table[Product[StirlingS1[n,k],{k,n}],{n,10}]] (* Harvey P. Dale, Oct 18 2014 *)

log(a(n)) ~ n^2 * (log(n) + Pi^2/6 - 3/2) / 2. - Vaclav Kotesovec, Feb 27 2021

a(11) from Harvey P. Dale, Oct 18 2014

A058808 Product{k=1 to n}[S(n,k)], where S(n,k) is a Stirling number of the second kind. (S(n,k) = number of ways of partitioning a set of n elements into k nonempty subsets.)

Original entry on oeis.org

1, 1, 3, 42, 3750, 2720250, 19512927000, 1631977354072800, 1833446251541145780000, 31323109023670061678062500000, 9087660958278168844264470405352500000

Offset: 1

Leroy Quet, Jan 02 2001

			a(4) = S(4,1)*S(4,2)*S(4,3)*S(4,4) = 1*7*6*1 = 42.

Vaclav Kotesovec, Table of n, a(n) for n = 1..40

Cf. A058807, A294373.

a:=n->mul(stirling2(n, k), k=1..n): seq(a(n), n=1..12); # Zerinvary Lajos, Jun 28 2007

Table[Product[StirlingS2[n, k], {k, 1, n}], {n, 1, 12}] (* Vaclav Kotesovec, Feb 26 2021 *)

a(n) = prod(k=1, n, stirling(n, k, 2)); \\ Michel Marcus, Dec 12 2015

log(a(n)) ~ n^2 * (log(n) + gamma - 3/2) / 2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 27 2021

A342170 Product of first n little Schröder numbers.

Original entry on oeis.org

1, 1, 3, 33, 1485, 292545, 264168135, 1130375449665, 23503896724884345, 2422053053602606867905, 1256704025339194996874320395, 3326147448057830199712191898815585, 45398150793225628820115544929795174823365, 3225056167710201318911738099365978237877235350145

Offset: 0

Vaclav Kotesovec, Mar 03 2021

nonn

Eric Weisstein's World of Mathematics, Super Catalan Number.
Wikipedia, Schroeder-Hipparchus numbers.

Cf. A001003, A003046, A003266, A070825, A294373, A342166.

b:= proc(n) option remember; `if`(n<2, 1,
      ((6*n-3)*b(n-1)-(n-2)*b(n-2))/(n+1))
    end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 03 2021

Table[Product[Hypergeometric2F1[1-k, k+2, 2, -1], {k, 1, n}], {n, 0, 15}]
FoldList[Times, 1, Table[Hypergeometric2F1[1 - n, n + 2, 2, -1], {n, 1, 15}]]

a(n) = Product_{k=1..n} A001003(k).

a(n) ~ c * (1 + sqrt(2))^(n*(n+2)) * exp(3*n/2) / (2^((7*n + 3)/4) * Pi^((2*n + 3)/4) * n^(3*n/2 + 3/2 + 9/(16*sqrt(2)))), where c = 0.89405100528141459535141257102427907468205556782800836208733677564241771912...

A342166 Product of first n Fubini numbers.

Original entry on oeis.org

1, 1, 3, 39, 2925, 1582425, 7410496275, 350464600333575, 191295845123076910125, 1355763582602823185129417625, 138623522325287867599380791765497875, 224935042709004795568466587349227029537282375, 6318777956744220129890735589019782971247629409914638125

Offset: 0

Vaclav Kotesovec, Mar 03 2021

nonn

Wikipedia, Ordered Bell number

Cf. A000670, A003046, A003266, A070825, A294373, A342170.

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n, j), j=1..n))
    end:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*g(n)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 03 2021

Table[Product[Sum[j!*StirlingS2[k, j], {j, 0, k}], {k, 1, n}], {n, 0, 12}]
Table[Product[PolyLog[-k, 1/2]/2, {k, 1, n}], {n, 0, 12}]
FoldList[Times, 1, Table[PolyLog[-n, 1/2]/2, {n, 1, 12}]]

a(n) = Product_{k=1..n} A000670(k).
a(n) ~ c * BarnesG(n+2) / (2^n * log(2)^(n*(n+3)/2)), where c = 0.960303470666951851619546415046950178638511457142008903473074598398282549...
a(n) ~ c * Pi^((n+1)/2) * n^(n^2/2 + n + 5/12) / (A * 2^((n-1)/2) * exp(3*n^2/4 + n - 1/12) * log(2)^(n*(n+3)/2)), where A is the Glaisher-Kinkelin constant A074962.

A343263 a(0) = 1; a(n+1) = exp(-a(n)) * Sum_{k>=0} a(n)^k * k^n / k!.

Original entry on oeis.org

1, 1, 1, 2, 22, 301554, 2493675105669492542968967478

Offset: 0

Ilya Gutkovskiy, Apr 09 2021

nonn

The next term is too large to include.

Cf. A000110, A003659, A067691, A110386, A189233, A242817, A292860, A294373.

a[0] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 1, k] a[n - 1]^k, {k, 0, n - 1}]; Table[a[n], {n, 0, 6}]
a[0] = 1; a[n_] := a[n] = BellB[n - 1, a[n - 1]]; Table[a[n], {n, 0, 6}]

a(0) = 1; a(n+1) = n! * [x^n] exp(a(n) * (exp(x) - 1)).

a(0) = 1; a(n+1) = Sum_{k=0..n} Stirling2(n,k) * a(n)^k.

cp's OEIS Frontend

A058807 a(n) = Product_{k=1..n} s(n,k), where s(n,k) is unsigned Stirling number of the first kind. (s(n,k) = number of permutations of n elements which contain exactly k cycles.)

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A058808 Product{k=1 to n}[S(n,k)], where S(n,k) is a Stirling number of the second kind. (S(n,k) = number of ways of partitioning a set of n elements into k nonempty subsets.)

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A342170 Product of first n little Schröder numbers.

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A342166 Product of first n Fubini numbers.

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A343263 a(0) = 1; a(n+1) = exp(-a(n)) * Sum_{k>=0} a(n)^k * k^n / k!.

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