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 1-7 of 7 results.

A383888 a(n) = Sum_{k=0..n} 3^k * binomial(n+k-1,k).

Original entry on oeis.org

1, 4, 34, 334, 3478, 37384, 409960, 4558306, 51199558, 579554056, 6600532684, 75546800476, 868224027916, 10012494936136, 115804853315332, 1342795688895754, 15604522381828678, 181690692393744376, 2119144763079629452, 24754486729805925124, 289563977079418497748
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A384950, A385438.
Cf. A064063, A385319.

Programs

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

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^2/(1-2*x) )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x) / (1+x)^2 ).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n,k).
a(n) = (-2)^(-n)*(1 - (-6)^n*binomial(2*n-1, n)*(hypergeom([1, 2*n], [1+n], 3) - 1)). - Stefano Spezia, Aug 02 2025
a(n) ~ 2^(2*n) * 3^(n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025

A385438 a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k-1,k).

Original entry on oeis.org

1, 10, 208, 4888, 121132, 3092950, 80506684, 2123780536, 56581885468, 1518936682888, 41021505946468, 1113273696074968, 30335161535834212, 829405495046080612, 22742967214283811976, 625193974445825554408, 17223870801864911429404, 475423918887141016417144
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A383888, A384950.
Cf. A385475.

Programs

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

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^4/(1-2*x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x)^3 / (1+x)^4 ). See A385475.
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n,k).
a(n) = (-2)^(-3*n) - 3^n*binomial(4*n-1, n)*(hypergeom([1, 4*n], [1+n], 3) - 1). - Stefano Spezia, Aug 02 2025
a(n) ~ 2^(8*n + 1/2) / (11 * 3^(2*n - 3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025

A386764 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(2*n+k-1,k).

Original entry on oeis.org

1, 13, 319, 8872, 260511, 7885793, 243404884, 7615561092, 240662849871, 7663737420223, 245529092332599, 7904950462600512, 255541233005365956, 8289112264436610828, 269663237466343607464, 8794852773491081069132, 287467221911677590185391, 9414259968096351504747483
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Crossrefs

Cf. A386763, A386765.
Cf. A384950, A386770.

Programs

  • PARI
    a(n) = sum(k=0, n, 5^k*3^(n-k)*binomial(2*n+k-1, k));

Formula

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
a(n) = [x^n] ( (1+3*x)^3/(1-2*x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+3*x)^3 ). See A386770.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n,k).
a(n) = (27/4)^n - 5^n*binomial(3*n-1, n)*(hypergeom([1, 3*n], [1+n], 5/3) - 1). - Stefano Spezia, Aug 02 2025

A385474 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+x)^3 ).

Original entry on oeis.org

1, 7, 76, 991, 14281, 219172, 3512440, 58096591, 984340003, 16996883887, 298017184048, 5291703108292, 94961611382860, 1719543577996888, 31379622840361696, 576519956457976495, 10655055147825932119, 197959348525977645781, 3695112941037246866044
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A064063, A385475.
Cf. A384950.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^2/(1+x)^3)/x)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(3*(n+1), k)*binomial(3*n-k+1, n-k))/(n+1);

Formula

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

A385667 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).

Original entry on oeis.org

1, 10, 151, 2542, 44983, 819160, 15197404, 285653350, 5421341311, 103659081034, 1993769491591, 38532753357064, 747680491747876, 14556620712375856, 284217498703106224, 5563106991308471062, 109124768598722692111, 2144648671343440349182
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Links

Crossrefs

Cf. A384950.

Programs

  • Magma
    [&+[2^(n-k) * Binomial(3*n+1,k) * Binomial(3*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 05 2025
  • Mathematica
    Table[Sum[2^(n-k)*Binomial[3*n+1,k]*Binomial[3*n-k,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 05 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+1, k)*binomial(3*n-k, n-k));

Formula

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

A386863 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 8, 117, 1948, 34283, 622272, 11519692, 216193460, 4098365799, 78293227384, 1504814127893, 29066030323920, 563717999500852, 10970568626688704, 214125123753359544, 4189892211091193380, 82166338354628744159, 1614453403457943056184, 31776198133079795063887
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2025

Keywords

Crossrefs

Cf. A384950, A386866.

Programs

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

Formula

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

A386866 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+2,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 9, 132, 2197, 38649, 701292, 12979360, 243541725, 4616122851, 88173726337, 1694554311888, 32728267058604, 634701136059532, 12351249029265816, 241061116082196072, 4716751239386395885, 92494719333403946583, 1817328001770278062299, 35768122814759119268788
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2025

Keywords

Crossrefs

Cf. A384950, A386863.
Cf. A386836.

Programs

  • Mathematica
    Table[Sum[2^(n-k) Binomial[3n+2,k]Binomial[3n-k-1,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+2, k)*binomial(3*n-k-1, n-k));

Formula

a(n) = [x^n] (1+x)^(3*n+2)/(1-2*x)^(2*n).
a(n) = [x^n] 1/((1-x)^3 * (1-3*x)^(2*n)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n+2,k) * binomial(n-k+2,n-k).
a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k-1,k) * binomial(n-k+2,n-k).
Showing 1-7 of 7 results.

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

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