A360990 -id:A360990 - cp's OEIS frontend

A360987 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^2).

1, 1, -3, -23, 233, 3521, -62171, -1416407, 35880977, 1095318721, -36224195059, -1387587617239, 56675849155705, 2612993427672577, -127090039302776395, -6852033608852338199, 386750643197222855969, 23875394847093826450049

Offset: 0

Seiichi Manyama, Feb 27 2023

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Sum_{k=0..n} (2*n - 2*k + 1)^(k-1) * (2*k)^(n-k) * binomial(n,k) = (2*n+1)^(n-1) = A052750(n). - Vaclav Kotesovec, Jul 03 2025

Seiichi Manyama, Table of n, a(n) for n = 0..367

Cf. A052750, A141369, A196198, A360988, A360989, A360990.

Cf. A168600.

a(n) = sum(k=0, n, (2*n-2*k+1)^(k-1)*(-2*k)^(n-k)*binomial(n, k));

a(n) = Sum_{k=0..n} (2*n - 2*k + 1)^(k-1) * (-2*k)^(n-k) * binomial(n,k).

a(0) = 1; a(n) = (-1)^(n-1) * (n-1)! * Sum_{i, j, k>=0 and i+j+k=n-1} (-1)^i * (n-i) * a(i) * a(j) * a(k)/(i! * j! * k!). - Seiichi Manyama, Jul 06 2025

A360988 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^3).

Original entry on oeis.org

1, 1, -5, -44, 829, 14656, -488897, -13063616, 629051449, 22531502080, -1420908901469, -63859764079616, 4983153798630709, 269501734545522688, -25073583375908431769, -1585437525801020801024, 171326697778165116452977, 12401692280007001315999744

Offset: 0

Seiichi Manyama, Feb 27 2023

sign

Cf. A141369, A196198, A360987, A360989, A360990.

Cf. A168601.

a(n) = sum(k=0, n, (3*n-3*k+1)^(k-1)*(-3*k)^(n-k)*binomial(n, k));

a(n) = Sum_{k=0..n} (3*n - 3*k + 1)^(k-1) * (-3*k)^(n-k) * binomial(n,k).

a(0) = 1; a(n) = (-1)^(n-1) * (n-1)! * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (-1)^i * (n-i) * a(i) * a(j) * a(k) * a(l)/(i! * j! * k! * l!). - Seiichi Manyama, Jul 06 2025

A360989 E.g.f. satisfies A(x) = exp(x / A(-x)^2).

Original entry on oeis.org

1, 1, 5, 1, -231, 81, 55453, -40431, -30313231, 33477985, 29630916981, -43713004191, -45378051616631, 83666428734513, 100216964952070541, -221570666935625999, -301515678925659598623, 777062158771833364929, 1185517627245415533666277

Offset: 0

Seiichi Manyama, Feb 27 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..366

Cf. A141369, A196198, A360987, A360988, A360990.

a(n) = sum(k=0, n, (-2*n+2*k+1)^(k-1)*(2*k)^(n-k)*binomial(n, k));

a(n) = Sum_{k=0..n} (-2*n + 2*k + 1)^(k-1) * (2*k)^(n-k) * binomial(n,k).

cp's OEIS Frontend

A360987 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^2).

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A360988 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^3).

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A360989 E.g.f. satisfies A(x) = exp(x / A(-x)^2).

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Author

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PARI

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