cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

1, 1, -3, -23, 233, 3521, -62171, -1416407, 35880977, 1095318721, -36224195059, -1387587617239, 56675849155705, 2612993427672577, -127090039302776395, -6852033608852338199, 386750643197222855969, 23875394847093826450049
Offset: 0

Views

Author

Seiichi Manyama, Feb 27 2023

Keywords

Comments

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

Links

Crossrefs

Cf. A052750, A141369, A196198, A360988, A360989, A360990.
Cf. A168600.

Programs

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

Formula

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

Views

Author

Seiichi Manyama, Feb 27 2023

Keywords

Crossrefs

Cf. A141369, A196198, A360987, A360989, A360990.
Cf. A168601.

Programs

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

Formula

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

A360990 E.g.f. satisfies A(x) = exp(x / A(-x)^3).

Original entry on oeis.org

1, 1, 7, -8, -827, 2896, 452179, -2511872, -560237303, 4254259456, 1237434920191, -11907540107264, -4275828959720435, 49800209789734912, 21288959122755516235, -290981680034059649024, -144324916601232035246831, 2264121148389579474141184
Offset: 0

Views

Author

Seiichi Manyama, Feb 27 2023

Keywords

Crossrefs

Cf. A141369, A196198, A360987, A360988, A360989.

Programs

  • Maple
    A360990 := proc(n)
    add((-3*n+3*k+1)^(k-1)*(3*k)^(n-k)*binomial(n,k),k=0..n) ;
end proc:
seq(A360990(n),n=0..60) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n, (-3*n+3*k+1)^(k-1)*(3*k)^(n-k)*binomial(n, k));

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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