A383380 -id:A383380 - cp's OEIS frontend

A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

1, 3, 14, 82, 576, 4688, 43264, 445632, 5062016, 62812288, 844863744, 12239474432, 189939644416, 3142842052608, 55223903596544, 1026805938614272, 20139224002953216, 415503046091767808, 8994794537935765504, 203848794955954716672, 4826475681472562855936, 119162892472107134353408

Offset: 0

Seiichi Manyama, Apr 24 2025

Cf. A001909, A383382, A383383, A383384.

Cf. A000023, A052124, A087981, A383380.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^5))

a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = 1, a(1) = 3; a(n) = (n+2)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A383378 Expansion of e.g.f. exp(-3*x) / (1-x)^4.

Original entry on oeis.org

1, 1, 5, 21, 129, 897, 7317, 67365, 692577, 7849953, 97199109, 1304688789, 18863836065, 292198665249, 4826470920021, 84669407740773, 1571901715253313, 30786460730863425, 634323280633460613, 13714611211502376597, 310448651226154786881, 7342298348439393120321

Offset: 0

Seiichi Manyama, Apr 24 2025

Column k=3 of A383341.
Cf. A000261, A383344, A383380.
Cf. A010843, A137775, A383382.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-3*x)/(1-x)^4))

a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * binomial(k+3,3)/(n-k)!.
a(0) = a(1) = 1; a(n) = n*a(n-1) + 3*(n-1)*a(n-2).
a(n) = A137775(n+2)/(3*(n+1)).
a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+3)). - Vaclav Kotesovec, Apr 25 2025

cp's OEIS Frontend

A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

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