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.

A381751 Expansion of exp( Sum_{k>=1} binomial(8*k-1,2*k-1) * x^k/k ).

Original entry on oeis.org

1, 7, 252, 12866, 767460, 50005591, 3449225652, 247579862356, 18301102679444, 1383742325041292, 106516121515030768, 8319491960857739258, 657680525420544788060, 52522142073165048614002, 4230907373618147894630904, 343379827862952363210331624, 28051180121294369965012932980
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(8*k-1, 2*k-1)*x^k/k)))

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(8*k-1,2*k-1) * a(n-k).

A381746 Expansion of exp( Sum_{k>=1} binomial(10*k-1,2*k) * x^k/k ).

Original entry on oeis.org

1, 36, 2586, 235884, 24284907, 2689924444, 312907382800, 37699275223260, 4663450108073401, 588854988193808392, 75589352418472567340, 9834912295258236849604, 1294095251234713917535805, 171909332777340128148714400, 23024035140764003881788203616
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(10*k-1, 2*k)*x^k/k)))

Formula

G.f. A(x) satisfies A(x^2) = B(x) * B(-x), where B(x) is the g.f. of A118971.
a(n) = Sum_{k=0..2*n} (-1)^k * A118971(k) * A118971(2*n-k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(10*k-1,2*k) * a(n-k).
G.f.: B(x)^4, where B(x) is the g.f. of A381752.

A381744 Expansion of exp( Sum_{k>=1} binomial(6*k-1,2*k) * x^k/k ).

Original entry on oeis.org

1, 10, 215, 5942, 186111, 6283192, 222992692, 8201608382, 309834609743, 11950890428170, 468707758663887, 18634632264615272, 749325132218313540, 30422303269317412048, 1245346665979469486376, 51343805279989437688334, 2130090659402456357279919, 88858984785475871013971710
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(6*k-1, 2*k)*x^k/k)))

Formula

G.f. A(x) satisfies A(x^2) = B(x) * B(-x), where B(x) is the g.f. of A006013.
a(n) = Sum_{k=0..2*n} (-1)^k * A006013(k) * A006013(2*n-k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(6*k-1,2*k) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A182960.
Showing 1-3 of 3 results.