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.

A379021 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^2 ).

Original entry on oeis.org

1, 4, 26, 206, 1813, 17032, 167287, 1697044, 17643322, 186997570, 2012973499, 21948003052, 241883091289, 2690117648372, 30153678822007, 340305271736134, 3863616751855069, 44097785533620550, 505692279260755753, 5823592506326814874, 67320958983831426221
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Links

Crossrefs

Cf. A007863, A379023, A379024.
Cf. A215654, A379022.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379022(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(3/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A215654.
a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..n} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-k).

A379026 a(n) = Sum_{k=0..n} binomial(4*n+k-1,k) * binomial(4*n+k,n-k).

Original entry on oeis.org

1, 8, 136, 2612, 52888, 1103248, 23458756, 505519792, 11001461560, 241240165796, 5321735043496, 117969960106380, 2625673485660100, 58638653062716488, 1313363972969179904, 29489827322243843032, 663600214363813934328, 14961465721142755457484
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Crossrefs

Cf. A262910, A379022, A379025.
Cf. A379024.

Programs

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

Formula

a(n) = [x^n] ( (1 + x)/(1 - x - x^2) )^(4*n).

A379023 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^3 ).

Original entry on oeis.org

1, 6, 57, 653, 8277, 111780, 1576671, 22955298, 342377304, 5204438258, 80334470136, 1255798641861, 19840021268937, 316286673287724, 5081503084814883, 82193597974971157, 1337397202150986387, 21875767255039745856, 359499909751084059372, 5932767953991599086905
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Links

Crossrefs

Cf. A007863, A379021, A379024.
Cf. A239107, A379025.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379025(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(4/3)) )^3.
(3) A(x) = B(x)^3 where B(x) is the g.f. of A239107.
a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(3*(n+1)).
a(n) = 3 * Sum_{k=0..n} binomial(3*n+k+3,k) * binomial(3*n+k+3,n-k)/(3*n+k+3) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*n+k+3,n-k).
Showing 1-3 of 3 results.

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