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.

A387310 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 21, 330, 4690, 63690, 844662, 11052496, 143462592, 1852852365, 23853938185, 306473670822, 3932435239278, 50417223635233, 646085510253645, 8277409340709240, 106037993391958936, 1358437551566242347, 17404555385537336583, 223025734596708637750, 2858460480570547144110
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A387309, A387311.

Programs

  • Magma
    [&+[3^k * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
  • Mathematica
    Table[Sum[3^k * Binomial[n+2,k+2]*Binomial[2*k+4, k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(n+2, k+2)*binomial(2*k+4, k+4));

Formula

n*(n+4)*a(n) = (n+2) * (7*(2*n+3)*a(n-1) - 13*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+2).
E.g.f.: exp(7*x) * BesselI(2, 6*x) / 9, with offset 2.

A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 14, 174, 2128, 26045, 320082, 3951493, 48987848, 609592347, 7610525650, 95287524332, 1196054790168, 15046318739803, 189654839753750, 2394743468261190, 30285593026553536, 383554551776056139, 4863775493104574634, 61748210178809072722, 784757334938247965840, 9983152795673915802399
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A387310, A387311.
Cf. A026376, A331793.
Cf. A385716.

Programs

  • Magma
    [&+[3^k * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025
  • Mathematica
    Table[Sum[3^k*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(n+1, k+1)*binomial(2*k+2, k+2));

Formula

G.f.: ((1-7*x)/sqrt((1-x) * (1-13*x)) - 1)/(18*x^2).
n*(n+2)*a(n) = (n+1) * (7*(2*n+1)*a(n-1) - 13*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+1).
E.g.f.: exp(7*x) * BesselI(1, 6*x) / 3, with offset 1.

A387316 Expansion of 1/((1-x) * (1-13*x))^(7/2).

Original entry on oeis.org

1, 49, 1498, 36750, 792246, 15681666, 292137846, 5201141946, 89399571261, 1494080348761, 24403114463728, 391038174645664, 6165638429715492, 95880046644705876, 1473241291627666488, 22401020288076984120, 337479336374849120991, 5042656883996693680719
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A385716, A387315.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-13*x))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-13*x))^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-13*x))^(7/2))

Formula

n*a(n) = (14*n+35)*a(n-1) - 13*(n+5)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 13^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = Sum_{k=0..n} (-12)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 12^k * 13^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387311(n).
a(n) = (-1)^n * Sum_{k=0..n} 14^k * (13/14)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).
Showing 1-3 of 3 results.

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