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 1-4 of 4 results.

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

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

A386829 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 13, 204, 3457, 61006, 1103598, 20299434, 377871297, 7097430726, 134243202358, 2553356761264, 48788507855562, 935791802540596, 18007015501848568, 347459946354962694, 6720599552926105377, 130263082422599127366, 2529516572366126192478
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386830, A386831.
Cf. A385669, A386763.

Programs

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

Formula

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

A386769 Expansion of (1/x) * Series_Reversion( x * (1-2*x) / (1+3*x)^2 ).

Original entry on oeis.org

1, 8, 89, 1162, 16646, 253218, 4016769, 65713602, 1100773166, 18786755128, 325518195674, 5711193510092, 101260078423336, 1811480526001238, 32657053453306929, 592701233703282882, 10820725155122336406, 198584549759713158048, 3661487133197990007534
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A064063, A386763.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*(n+1),k) * binomial(2*n-k,n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+3*x)^2 / (1-2*x) )^(n+1).
D-finite with recurrence 2*(n+1)*a(n) +(-31*n+29)*a(n-1) +90*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 03 2025
Showing 1-4 of 4 results.

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