A380212 -id:A380212 - cp's OEIS frontend

A380213 Expansion of e.g.f. exp( 1/(1-2*x)^(5/2) - 1 ).

1, 5, 60, 965, 19315, 459420, 12597775, 389902175, 13410470700, 506509866575, 20811096098725, 923085833362500, 43921261488000625, 2229827043134538125, 120239258292160027500, 6859351794101350278125, 412554191158956599261875, 26080572238227541202917500

Offset: 0

Seiichi Manyama, Jan 16 2025

nonn

Cf. A049118, A380212.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1/(1-2*x)^(5/2)-1)))

a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * |Stirling1(n,k)| * Bell(k).

a(n) = (1/e) * (-2)^n * n! * Sum_{k>=0} binomial(-5*k/2,n)/k!.

A380259 Expansion of e.g.f. exp( (1/(1-2*x)^(3/2) - 1)/3 ).

Original entry on oeis.org

1, 1, 6, 51, 561, 7566, 120711, 2221311, 46269126, 1075249881, 27560477331, 771948530046, 23446574573841, 767288588019201, 26905482997736526, 1006166248423254171, 39962774633459923881, 1679677496419394133846, 74471142324541556576151

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049377, A380212.

Table[Sum[3^k * 2^(n-k) * Abs[StirlingS1[n,k]] * BellB[k, 1/3], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-2*x)^(3/2)-1)/3)))

a(n) = Sum_{k=0..n} 2^(n-k) * |Stirling1(n,k)| * A004212(k) = Sum_{k=0..n} 3^k * 2^(n-k) * |Stirling1(n,k)| * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/3)) * (-2)^n * n! * Sum_{k>=0} binomial(-3*k/2,n)/(3^k * k!).
a(n) ~ 2^(n + 3/10) * n^(n - 1/5) * exp(-1/3 + 2^(1/5)*n^(1/5)/4 + 5*2^(3/5)*n^(3/5)/6 - n) / sqrt(5) * (1 + 2^(4/5) / (30 * n^(1/5))). - Vaclav Kotesovec, Jan 23 2025

cp's OEIS Frontend

A380213 Expansion of e.g.f. exp( 1/(1-2*x)^(5/2) - 1 ).

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