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.

A386843 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 6, 39, 268, 1905, 13842, 102123, 761880, 5732325, 43417630, 330620895, 2528772132, 19412942809, 149497184298, 1154365194195, 8934458916912, 69291946278861, 538372925816886, 4189702003359687, 32651982699233340, 254800541773725633, 1990683254889381954
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386844, A386845.
Cf. A059304, A178792.
Cf. A116881.

Programs

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

Formula

a(n) = [x^n] (1+x)^(2*n+2)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(2*n+2,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(n+k,k).

A386845 a(n) = Sum_{k=0..n} binomial(4*n+2,k) * binomial(4*n-k,n-k).

Original entry on oeis.org

1, 10, 143, 2264, 37601, 642086, 11165395, 196658228, 3496849349, 62636490818, 1128525823927, 20429545554000, 371294468833193, 6770529284259934, 123811606398566299, 2269695135303598188, 41697091253148057485, 767476182916622450810, 14149874243880085356415
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386843, A386844.

Programs

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

Formula

a(n) = [x^n] (1+x)^(4*n+2)/(1-x)^(3*n+1).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(4*n+2,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(3*n+k,k).

A386869 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+2,k) * binomial(3*n-k,n-k).

Original entry on oeis.org

1, 11, 168, 2839, 50333, 917604, 17036260, 320383295, 6082829067, 116342007859, 2238247173440, 43266114873636, 839661737871388, 16349646755219432, 319263686177979564, 6249714381417109903, 122603983720769666087, 2409746305286188995681
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2025

Keywords

Crossrefs

Cf. A385667, A386844.

Programs

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

Formula

a(n) = [x^n] (1+x)^(3*n+2)/(1-2*x)^(2*n+1).
a(n) = [x^n] 1/((1-x)^2 * (1-3*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * (n-k+1) * binomial(3*n+2,k).
a(n) = Sum_{k=0..n} 3^k * (n-k+1) * binomial(2*n+k,k).
D-finite with recurrence 544*n*(2*n-1)*a(n) +8*(618*n^2-9184*n+8025)*a(n-1) +2*(-276538*n^2+1112059*n-1061145)*a(n-2) +15327*(3*n-4)*(3*n-5)*a(n-3)=0. - R. J. Mathar, Aug 19 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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