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.

A386763 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(n+k-1,k).

Original entry on oeis.org

1, 8, 114, 1862, 32246, 576768, 10529544, 194960802, 3647285766, 68772760928, 1304858513324, 24882531221292, 476462691535436, 9155397868559288, 176447193966483204, 3409285356643013082, 66020593061854488006, 1280989373915746600848, 24897996624141835608684
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Crossrefs

Cf. A386764, A386765.
Cf. A383888, A386769.

Programs

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

Formula

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] ( (1+3*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+3*x)^2 ). See A386769.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(2*n,k).
a(n) = (-9/2)^n*(1 - (-10/9)^n*binomial(2*n-1, n)*(hypergeom([1, 2*n], [1+n], 5/3) - 1)). - Stefano Spezia, Aug 02 2025

A386830 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).

Original entry on oeis.org

1, 18, 459, 12942, 382671, 11632248, 360048924, 11287595862, 357239123631, 11389281564978, 365227235524539, 11767662196724232, 380651590433357316, 12354006908520865008, 402088229127633026304, 13119017347331737771302, 428955765661154879370351
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386829, A386831.
Cf. A386764.

Programs

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

Formula

a(n) = [x^n] (1+3*x)^(3*n+1)/(1-2*x)^(2*n+1).
a(n) = [x^n] 1/((1-3*x) * (1-5*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(2*n+k,k).
Conjecture D-finite with recurrence +8*n*(2*n-3)*a(n) +6*(-108*n^2+207*n-80)*a(n-1) +405*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Aug 19 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

A386770 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+3*x)^3 ).

Original entry on oeis.org

1, 13, 244, 5397, 130961, 3372268, 90497184, 2503434117, 70883043571, 2044268649573, 59842331451024, 1773506049794412, 53107658756034156, 1604418047921589928, 48841208603255888264, 1496711470907670605157, 46134317696761847385591, 1429405788411234205692583
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A385474, A386764.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^2/(1+3*x)^3)/x)
  • PARI
    a(n) = sum(k=0, n, 3^k*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} 3^k * 2^(n-k) * binomial(3*(n+1),k) * binomial(3*n-k+1,n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+3*x)^3 / (1-2*x)^2 )^(n+1).
D-finite with recurrence 32*(n+1)*(2*n+1)*a(n) +48*(-81*n^2+27*n-7)*a(n-1) +162*(414*n^2-891*n+605)*a(n-2) -32805*(3*n-4)*(3*n-5)*a(n-3)=0. - R. J. Mathar, Aug 03 2025
Showing 1-4 of 4 results.

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