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 1-5 of 5 results.

A385716 Expansion of 1/((1-x) * (1-13*x))^(3/2).

Original entry on oeis.org

1, 21, 348, 5320, 78135, 1120287, 15805972, 220445316, 3047961735, 41857891075, 571725145992, 7774356136092, 105324231178621, 1422411298153125, 19157947746089520, 257427540725705056, 3451990965984505251, 46205867184493459023, 617482101788090727220, 8239952016851603641320
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A331516, A385563.
Cf. A340973, A386362.

Programs

  • Mathematica
    Module[{x}, CoefficientList[Series[1/((1-x)*(1-13*x))^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-13*x))^(3/2))

Formula

n*a(n) = (14*n+7)*a(n-1) - 13*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 3^k * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A386362(n).
a(n) = ((n+2)/2) * 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) = Sum_{k=0..n} (7/2)^k * (-13/14)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(3*n) * 13^(n + 3/2) / (36*sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

A385728 Expansion of 1/((1-2*x) * (1-6*x))^(3/2).

Original entry on oeis.org

1, 12, 102, 760, 5310, 35784, 235788, 1530288, 9824310, 62557000, 395797908, 2491381776, 15616141996, 97537784400, 607391245080, 3772617319008, 23379854507046, 144605546475336, 892834113930180, 5504041611527760, 33883431379007364, 208327771987901808
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A385563, A385813.
Cf. A005572, A081671, A331515.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1 / ((1 - 2*x) * (1 - 6*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 22 2025
  • Mathematica
    Module[{a, n}, RecurrenceTable[{a[n] == ((8*n+4)*a[n-1] - 12*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 12}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)
CoefficientList[Series[ 1/((1-2*x)*(1-6*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 22 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(3/2))

Formula

n*a(n) = (8*n+4)*a(n-1) - 12*(n+1)*a(n-2) for n > 1.
a(n) = (1/2)^n * Sum_{k=0..n} 3^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A005572(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} 2^k * (-3/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(n) * 2^(n - 1/2) * 3^(n + 3/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 21 2025

A385813 Expansion of 1/((1-3*x) * (1-7*x))^(3/2).

Original entry on oeis.org

1, 15, 156, 1400, 11655, 92925, 721140, 5496300, 41361255, 308344025, 2282167272, 16795140180, 123030071437, 897791417775, 6530377362480, 47370038320800, 342794475282915, 2475479922896925, 17843821672113780, 128412824128709400, 922775179449162501, 6622378039719342615
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A385563, A385728.
Cf. A098409, A182401.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 25 2025
  • Mathematica
    CoefficientList[Series[1/((1-3x)*(1-7*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 25 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(3/2))

Formula

n*a(n) = (10*n+5)*a(n-1) - 21*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A182401(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (5/2)^k * (-21/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).

A387237 Expansion of 1/((1-x) * (1-5*x))^(5/2).

Original entry on oeis.org

1, 15, 145, 1155, 8260, 55188, 351960, 2170080, 13042095, 76827465, 445335891, 2547479025, 14412134100, 80773641900, 449065521300, 2479190589180, 13603361708775, 74238475926825, 403197150223175, 2180369322394725, 11744998515662720, 63044308615576200, 337323759106291100
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A026375, A385563, A387238.
Cf. A026377, A374508.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-5*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-5*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(5/2))

Formula

n*a(n) = (6*n+9)*a(n-1) - 5*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k) = (binomial(n+4,2)/6) * A026377(n+2).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).

A387238 Expansion of 1/((1-x) * (1-5*x))^(7/2).

Original entry on oeis.org

1, 21, 266, 2646, 22806, 178794, 1310694, 9140274, 61330269, 399107709, 2533330800, 15751925280, 96257031780, 579556206180, 3445117599480, 20252115155160, 117890464642335, 680320688005035, 3895668955041710, 22152779612619810, 125183331416173030
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A026375, A385563, A387237.
Cf. A126190, A374509, A387239.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-5*x))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-5*x))^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(7/2))

Formula

n*a(n) = (6*n+15)*a(n-1) - 5*(n+5)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387239(n).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).
Showing 1-5 of 5 results.

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