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

A168601 E.g.f. A(x) satisfies A(x) = exp( x*A(2*x)^3 ).

Original entry on oeis.org

1, 1, 13, 577, 65737, 17553361, 10704000277, 14796719274961, 46078793355045073, 320622833897384770657, 4940987529761176087132381, 167239138427197351352909547169, 12340269209872740741602975099855641
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Examples

			E.g.f: A(x) = 1 + x + 13*x^2/2! + 577*x^3/3! + 65737*x^4/4! +...

Crossrefs

Cf. A096538, A168600.
Cf. A360988.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(x*subst(A, x, 2*x)^3) ); n!*polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * (n-1)! * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (n-i)/2^i * a(i) * a(j) * a(k) * a(l)/(i! * j! * k! * l!). - Seiichi Manyama, Jul 06 2025

A385650 E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(2*x))^2 ).

Original entry on oeis.org

1, 4, 112, 8800, 1586944, 624664064, 536747751424, 1018102925488128, 4288756843049058304, 40076190507961751044096, 826422665125748814526283776, 37363126329930414708850363990016, 3679235193626553722088195031035805696, 784317990902751658071943156321585144528896
Offset: 0

Views

Author

Seiichi Manyama, Jul 06 2025

Keywords

Crossrefs

Cf. A385619, A385651.
Cf. A168600, A385648.

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{i, j, k>=0 and i+j+k=n-1} (n-i) * (2^j+1) * (2^k+1) * a(i) * a(j) * a(k)/(i! * j! * 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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