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.

A387276 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 20, 255, 2650, 24521, 210840, 1725234, 13631700, 104993955, 793367300, 5907885412, 43495473840, 317355930255, 2298888740400, 16555878011448, 118661449810320, 847132614218907, 6027874235210700, 42773816956415055, 302816249208061050, 2139537520524710691
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A387275, A387277, A387278.
Cf. A387239, A387273.

Programs

  • Magma
    [&+[3^(n-k) * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[3^(n-k)*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));

Formula

n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 21*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+5*x+x^2)^(n+3).
E.g.f.: exp(5*x) * BesselI(3, 2*x), with offset 3.

A387275 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 15, 154, 1350, 10890, 83650, 623056, 4547520, 32735085, 233369675, 1652203542, 11638252730, 81674873553, 571575363975, 3991529920440, 27829484027400, 193791573179883, 1348196149698885, 9372495529924710, 65120144658997050, 452263192928596896
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A387276, A387277, A387278.

Programs

  • Magma
    [&+[3^(n-k) * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[3^(n-k)*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));

Formula

n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 21*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 2*x), with offset 2.
a(n) ~ 7^(n + 5/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 31 2025

A387278 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 10, 78, 560, 3885, 26550, 180285, 1221400, 8272251, 56062550, 380361212, 2583867720, 17575724491, 119705522370, 816297170310, 5572945684800, 38088275031435, 260576833989150, 1784382167211378, 12229792774162800, 83888652677196591, 575858959975595010
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A098409, A387275, A387276, A387277.
Cf. A026376, A344055.

Programs

  • Magma
    [&+[3^(n-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^(n-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^(n-k)*binomial(n+1, k+1)*binomial(2*k+2, k+2));

Formula

n*(n+2)*a(n) = (n+1) * (5*(2*n+1)*a(n-1) - 21*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+5*x+x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 2*x), with offset 1.

A387285 Expansion of 1/((1-3*x) * (1-7*x))^(9/2).

Original entry on oeis.org

1, 45, 1143, 21615, 339273, 4678245, 58635291, 683043075, 7509979620, 78804997700, 795761196516, 7781553124020, 74052668067768, 688495224545640, 6273531262685808, 56167899323785920, 495164048016313071, 4305882857891533875, 36989093846561967825, 314291267108824487625
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A387277.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(9/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 26 2025
  • Mathematica
    Module[{x}, CoefficientList[Series[1/((1 - 3*x)*(1 - 7*x))^(9/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(9/2))

Formula

n*a(n) = (10*n+35)*a(n-1) - 21*(n+7)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 7^k * 3^(n-k) * binomial(-9/2,k) * binomial(-9/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * 3^(n-k) * binomial(-9/2,k) * binomial(n+8,n-k).
a(n) = Sum_{k=0..n} 4^k * 7^(n-k) * binomial(-9/2,k) * binomial(n+8,n-k).
a(n) = (binomial(n+8,4)/70) * A387277(n).
a(n) = (-1)^n * Sum_{k=0..n} 10^k * (21/10)^(n-k) * binomial(-9/2,k) * binomial(k,n-k).
Showing 1-4 of 4 results.

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