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 1-6 of 6 results.

A384950 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k-1,k).

Original entry on oeis.org

1, 7, 103, 1720, 30319, 550867, 10204660, 191606380, 3633593071, 69434167357, 1334845289023, 25787841299392, 500217562201348, 9736067678711524, 190051513661403112, 3719197868485767940, 72942019051301120239, 1433317465944902210161, 28212929859612197439829
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A383888, A385438.
Cf. A385474.

Programs

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

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
a(n) = [x^n] ( (1+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+x)^3 ). See A385474.
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k).
a(n) = 4^(-n) - 3^n*binomial(3*n-1, n)*(hypergeom([1, 3*n], [1+n], 3) - 1). - Stefano Spezia, Aug 02 2025
a(n) ~ 3^(4*n + 3/2) / (2^(2*n+3) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025

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

A385668 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

Original entry on oeis.org

1, 13, 274, 6466, 160564, 4104733, 106927384, 2822352952, 75224906716, 2020064928916, 54569506803574, 1481263780787122, 40369492671395476, 1103922337550185894, 30274295947104877312, 832318570941153758356, 22932288741241396871068, 633044952458953424442364
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Links

Crossrefs

Cf. A385438.

Programs

  • Magma
    [&+[2^(n-k) * Binomial(4*n+1,k) * Binomial(4*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 05 2025
  • Mathematica
    Table[Sum[2^(n-k)*Binomial[4*n+1,k]*Binomial[4*n-k,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, Aug 05 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));

Formula

a(n) = [x^n] (1+x)^(4*n+1)/(1-2*x)^(3*n+1).
a(n) = [x^n] 1/((1-x) * (1-3*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k,k).
a(n) ~ 2^(8*n + 5/2) / (11 * sqrt(Pi*n) * 3^(2*n - 1/2)). - Vaclav Kotesovec, Aug 05 2025

A386765 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(3*n+k-1,k).

Original entry on oeis.org

1, 18, 624, 24432, 1008876, 42927318, 1862060124, 81862383432, 3634739070876, 162615605774568, 7319222860673124, 331046648931192432, 15033834910528707876, 685059700337659528068, 31307482174782491223624, 1434354449577159551751432, 65858845473746133806094876
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Crossrefs

Cf. A386763, A386764.
Cf. A385438, A386771.

Programs

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

Formula

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
a(n) = [x^n] ( (1+3*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+3*x)^4 ). See A386771.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(4*n,k).
a(n) = (-2)^(-3*n)*81^n - 5^n*binomial(4*n - 1, n)*(hypergeom([1, 4*n], [1+n], 5/3) - 1). - Stefano Spezia, Aug 03 2025

A386864 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 11, 228, 5350, 132476, 3380955, 87974188, 2320223552, 61804459260, 1658904186124, 44796539697968, 1215611557398534, 33121179085639252, 905520072985022570, 24828701435772435528, 682496748843439692868, 18801742541632193099996, 518957827806105486222372
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2025

Keywords

Crossrefs

Cf. A385438, A386867.

Programs

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

Formula

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

A386867 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+2,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 12, 249, 5842, 144636, 3690840, 96028606, 2532467934, 67454242092, 1810467982144, 48887478311673, 1326582594222918, 36143786784056716, 988134308856642048, 27093384379207568028, 744735869371387679158, 20516019688758402141372, 566266846186568482197840
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2025

Keywords

Crossrefs

Cf. A385438, A386864.
Cf. A386837.

Programs

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

Formula

a(n) = [x^n] (1+x)^(4*n+2)/(1-2*x)^(3*n).
a(n) = [x^n] 1/((1-x)^3 * (1-3*x)^(3*n)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n+2,k) * binomial(n-k+2,n-k).
a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k-1,k) * binomial(n-k+2,n-k).
Showing 1-6 of 6 results.

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