A094418 -id:A094418 - cp's OEIS frontend

A384305 Expansion of Product_{k>=1} 1/(1 - k*x)^((5/6)^k).

1, 30, 615, 11260, 205695, 4013406, 88035585, 2255192280, 68859250020, 2506898720040, 107238427737876, 5281094776037040, 293625956135692020, 18139856902224931080, 1229886945212115522060, 90641666662687182976896, 7206758883035555464430370, 614391718014749017022916060

Offset: 0

Seiichi Manyama, May 26 2025

nonn

Cf. A084785, A384324, A384325, A384326.

Cf. A090358, A090362, A094418.

terms = 20; A[] = 1; Do[A[x] = -5*A[x] + 6*A[x/(1-x)]^(5/6) / (1-x)^5 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(exp(6*sum(k=1, N, sum(j=0, k, 5^j*j!*stirling(k, j, 2))*x^k/k)))

G.f. A(x) satisfies A(x) = A(x/(1-x))^(5/6) / (1-x)^5.
G.f.: exp(6 * Sum_{k>=1} A094418(k) * x^k/k).
G.f.: B(x)^30, where B(x) is the g.f. of A090358.
a(n) ~ (n-1)! / log(6/5)^(n+1). - Vaclav Kotesovec, May 31 2025

A238466 Generalized ordered Bell numbers Bo(9,n).

Original entry on oeis.org

1, 9, 171, 4869, 184851, 8772309, 499559571, 33190014069, 2520110222451, 215270320769109, 20431783142389971, 2133148392099721269, 242954208655633344051, 29977118969127060357909, 3983272698956314883956371, 567091857051921058649396469

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 9 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(10 - 9*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(10 - 9 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(10 - 9*exp(x)).
a(n) ~ n! / (10*(log(10/9))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 9*a(n-1) - 10*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A238467 Generalized ordered Bell numbers Bo(10,n).

Original entry on oeis.org

1, 10, 210, 6610, 277410, 14553010, 916146210, 67285818610, 5647734061410, 533307215001010, 55954905981282210, 6457903731351210610, 813080459351919805410, 110901542660769629769010, 16290196917457939734258210

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 10 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(11 - 10*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(11 - 10 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(11 - 10*exp(x)).
a(n) ~ n! / (11*(log(11/10))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 10*a(n-1) - 11*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

cp's OEIS Frontend

A384305 Expansion of Product_{k>=1} 1/(1 - k*x)^((5/6)^k).

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A238466 Generalized ordered Bell numbers Bo(9,n).

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A238467 Generalized ordered Bell numbers Bo(10,n).

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Author

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Magma

Mathematica

Formula