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.

A366695 G.f. satisfies A(x) = (1 + x)^3 + x*A(x)^2.

Original entry on oeis.org

1, 4, 11, 39, 166, 765, 3716, 18725, 96956, 512690, 2756806, 15027651, 82853678, 461215414, 2588619402, 14632777719, 83232244238, 476040155118, 2736005962314, 15793863291792, 91530881427964, 532343678619778, 3106141476531628, 18177446846299299
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*(k+1), n-k)*binomial(2*k, k)/(k+1));

Formula

G.f.: A(x) = 2*(1+x)^3 / (1+sqrt(1-4*x*(1+x)^3)).
a(n) = Sum_{k=0..n} binomial(3*(k+1),n-k) * binomial(2*k,k)/(k+1).

A381938 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 9, 52, 380, 3066, 26304, 235314, 2170312, 20487963, 196988392, 1922327792, 18990571724, 189548947601, 1908604524752, 19364096602370, 197761735366804, 2031444188437719, 20974821788118024, 217561484977675026, 2265961977605950416, 23688432825547509283
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(2*k+2, n-k)/(4*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(2*k+2,n-k)/(4*k+1).
a(n) = A381785(n) + A381785(n-1).

A381941 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 3, 10, 71, 644, 6461, 68971, 768054, 8820281, 103694479, 1241799996, 15095075897, 185769856443, 2310006893997, 28978952155943, 366315306556482, 4661272734504606, 59659914501348239, 767539555514812321, 9920124234695256009, 128744011085858468131, 1677087982747514335025
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(2*k+2, n-k)/(5*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(2*k+2,n-k)/(5*k+1).
Showing 1-3 of 3 results.