A380261 -id:A380261 - cp's OEIS frontend

A380215 Expansion of e.g.f. exp( (1+3*x)^(2/3) - 1 ).

1, 2, 2, 4, -12, 152, -2056, 34064, -663792, 14890656, -378083936, 10721383488, -335898007232, 11523599785856, -429685396446848, 17303743585216768, -748494039183318784, 34612915914568045056, -1704065501541830102528, 88989595986614229074944, -4913365756826406035999744

Offset: 0

Seiichi Manyama, Jan 16 2025

sign

Cf. A380208, A380261.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1+3*x)^(2/3)-1)))

a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * Stirling1(n,k) * Bell(k).
a(n) = (1/e) * 3^n * n! * Sum_{k>=0} binomial(2*k/3,n)/k!.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380261.
a(n) ~ (-1)^(n+1) * 2^(3/2) * sqrt(Pi) * 3^(n-1) * n^(n - 7/6) / (Gamma(1/3) * exp(n+1)). - Vaclav Kotesovec, Jan 19 2025

A380262 Expansion of e.g.f. exp( ((1+5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, -2, 16, -206, 3682, -84236, 2348704, -77241380, 2926735516, -125540336024, 6013069027648, -318093606114536, 18418565715581656, -1158626159228481488, 78679416565851286144, -5736477278907382585328, 446936684375920051751440, -37056888825921886749507872

Offset: 0

Seiichi Manyama, Jan 18 2025

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000085, A002119, A380261.

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

a(n) = Sum_{k=0..n} 5^(n-k) * Stirling1(n,k) * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * Stirling1(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

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

cp's OEIS Frontend

A380215 Expansion of e.g.f. exp( (1+3*x)^(2/3) - 1 ).

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