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 1-6 of 6 results.

A385563 Expansion of 1/((1-x) * (1-5*x))^(3/2).

Original entry on oeis.org

1, 9, 60, 360, 2055, 11403, 62132, 334260, 1781415, 9425295, 49581576, 259601004, 1353939405, 7038232425, 36484340400, 188665670880, 973545780195, 5014258620075, 25783103206100, 132378800689800, 678768332410245, 3476164133573505, 17782899991147500
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Partial sums of A383254.
Cf. A331516, A385716.
Cf. A385728, A385813.
Cf. A002212, A383949.

Programs

  • Mathematica
    Module[{a, n}, RecurrenceTable[{a[n] == ((6*n+3)*a[n-1] - 5*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 9}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(3/2))

Formula

n*a(n) = (6*n+3)*a(n-1) - 5*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^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*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A002212(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (3/2)^k * (-5/6)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(n) * 5^(n + 3/2) / (4*sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

A386362 Expansion of (1/x) * Series_Reversion( x/(1+7*x+9*x^2) ).

Original entry on oeis.org

1, 7, 58, 532, 5209, 53347, 564499, 6123481, 67732483, 761052565, 8662502212, 99671232514, 1157409133831, 13546774268125, 159649564550746, 1892849564159596, 22562032457415067, 270209749616920813, 3249905798884688038, 39237866746912398292, 475388228365424562019
Offset: 0

Views

Author

Seiichi Manyama, Aug 20 2025

Keywords

Crossrefs

Column k=3 of A386408.
Cf. A002212, A127846, A386389.
Cf. A000108, A337167, A385716.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+7*x+9*x^2))/x)

Formula

G.f.: 2/(1 - 7*x + sqrt((1-x) * (1-13*x))).
a(n) = (A337167(n+1) - A337167(n))/3.
(n+2)*a(n) = 7*(2*n+1)*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*k) * Catalan(k).
a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * Catalan(k+1).

A387210 Expansion of sqrt((1-x) / (1-13*x)^3).

Original entry on oeis.org

1, 19, 307, 4645, 67843, 969337, 13643533, 189953659, 2622877075, 35982412921, 491057325577, 6672763735183, 90347244052429, 1219537191931975, 16418449380961891, 220534056531679141, 2956293832279184659, 39559312793250153577, 528522358385088314425, 7051193680459915645903
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2025

Keywords

Links

Crossrefs

Cf. A163869, A163872, A387208.
Cf. A385716.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := Sqrt(((1-x) / (1-13*x)^3)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025
  • Mathematica
    CoefficientList[Series[Sqrt[(1-x)/(1-13*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sqrt((1-x)/(1-13*x)^3))

Formula

n*a(n) = (14*n+5)*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) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 3^k * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
a(n) ~ 4 * sqrt(3*n) * 13^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 23 2025

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.

A387315 Expansion of 1/((1-x) * (1-13*x))^(5/2).

Original entry on oeis.org

1, 35, 825, 16415, 297220, 5067972, 82893720, 1315073760, 20381376015, 310101196405, 4648184007467, 68817616687365, 1008344472704660, 14644604899082620, 211073938188085620, 3022082811670829676, 43017189132931007655, 609159438493806780405, 8586490781973282553375
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A385716, A387316.
Cf. A387230.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-13*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-13*x))^(5/2),{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))^(5/2))

Formula

n*a(n) = (14*n+21)*a(n-1) - 13*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 13^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-12)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 12^k * 13^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * A387310(n).
a(n) = (-1)^n * Sum_{k=0..n} 14^k * (13/14)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).

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 1-6 of 6 results.

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