A383341 -id:A383341 - cp's OEIS frontend

A383383 Expansion of e.g.f. exp(-4*x) / (1-x)^5.

1, 1, 6, 26, 176, 1296, 11296, 110176, 1197696, 14304896, 186166016, 2620022016, 39631568896, 640971452416, 11034441916416, 201411030081536, 3884642996289536, 78929236862140416, 1684881987266215936, 37695662812132212736, 881964287274876665856, 21536903057742987001856

Offset: 0

Seiichi Manyama, Apr 24 2025

Column k=4 of A383341.

Cf. A001909, A383381, A383382, A383384.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-4*x)/(1-x)^5))

a(n) = n! * Sum_{k=0..n} (-4)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = a(1) = 1; a(n) = n*a(n-1) + 4*(n-1)*a(n-2).
a(n) = A383344(n+2)/(4*(n+1)).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+4)). - 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

A383379 a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k,n)/(n-k)!.

Original entry on oeis.org

1, 1, 4, 21, 176, 1765, 22464, 331177, 5692672, 110286441, 2394828800, 57389046781, 1507401363456, 43018690418509, 1326170009092096, 43905977120300625, 1553942522589937664, 58544111242378404433, 2339326913228257886208, 98816004834223734304741

Offset: 0

Seiichi Manyama, Apr 24 2025

nonn

Main diagonal of A383341.

Cf. A295182.

a(n) = n!*sum(k=0, n, (-n)^(n-k)*binomial(n+k, k)/(n-k)!);

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

a(n) ~ phi^(3*n + 3/2) * n^n / (5^(1/4) * exp(phi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 25 2025

cp's OEIS Frontend

A383383 Expansion of e.g.f. exp(-4*x) / (1-x)^5.

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

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Author

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PARI

Formula

A383379 a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k,n)/(n-k)!.

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PARI

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