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.

A381752 Expansion of exp( Sum_{k>=1} binomial(10*k-1,2*k-1) * x^k/k ).

Original entry on oeis.org

1, 9, 525, 44067, 4338765, 467396050, 53346810991, 6339179481480, 775994115988525, 97182642466115275, 12392633418043399130, 1603634650155295053250, 210047857493659698690575, 27795006677556725604853840, 3710220786174094422360657000, 498998879378383167317202612400
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(10*k-1, 2*k-1)*x^k/k)))

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(10*k-1,2*k-1) * a(n-k).

A381745 Expansion of exp( Sum_{k>=1} binomial(8*k-1,2*k) * x^k/k ).

Original entry on oeis.org

1, 21, 903, 49525, 3070308, 204928371, 14369906538, 1043861319189, 77866470852108, 5929621690613108, 459076176165983247, 36026517938705145267, 2859318461620989381900, 229114879928544260792946, 18509862380800289696106372, 1506048000721264678984095445, 123303480420582227597300406588
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(8*k-1, 2*k)*x^k/k)))

Formula

G.f. A(x) satisfies A(x^2) = B(x)/x * B(-x)/(-x), where B(x) is the g.f. of A006632.
a(n) = Sum_{k=0..2*n} (-1)^k * A006632(k+1) * A006632(2*n-k+1).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(8*k-1,2*k) * a(n-k).
G.f.: B(x)^3, where B(x) is the g.f. of A381751.

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.