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 1-5 of 5 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.

A381753 Expansion of exp( Sum_{k>=1} binomial(5*k-1,2*k-1) * x^k/k ).

Original entry on oeis.org

1, 4, 50, 846, 16495, 349240, 7803823, 181135830, 4324897697, 105543188190, 2620784850325, 66005699547352, 1682046970846570, 43291586055360034, 1123707191010320955, 29382536610737191930, 773229801368332554273, 20463493681189771623960
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(5*k-1, 2*k-1)*x^k/k)))
    
  • PARI
    a(n) = 2*sum(k=0, n, binomial(5*n+2*k+2, k)*binomial(5*n+2, n-k)/(5*n+2*k+2));

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(5*k-1,2*k-1) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A060941.
a(n) = 2 * Sum_{k=0..n} binomial(5*n+2*k+2,k) * binomial(5*n+2,n-k)/(5*n+2*k+2).

A381757 Expansion of exp( Sum_{k>=1} binomial(7*k-1,2*k-1) * x^k/k ).

Original entry on oeis.org

1, 6, 161, 6062, 265868, 12720904, 643915209, 33905228350, 1838102210977, 101910583801012, 5751779249830131, 329359930638541776, 19087504000780665541, 1117418973753045781944, 65982722733895652916539, 3925378032146863676341770, 235048328495265879957413946
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(7*k-1, 2*k-1)*x^k/k)))

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(7*k-1,2*k-1) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A300386.

A381758 Expansion of exp( Sum_{k>=1} binomial(9*k-1,2*k-1) * x^k/k ).

Original entry on oeis.org

1, 8, 372, 24732, 1925394, 163883548, 14773987638, 1386341339430, 133994232166575, 13248555929274096, 1333732204895318366, 136243562694021684648, 14087033746990654649067, 1471456489458490198994856, 155042502964505871862313879, 16459391575059417875255359878
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(9*k-1, 2*k-1)*x^k/k)))

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(9*k-1,2*k-1) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A300387.
Showing 1-5 of 5 results.