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.

A360133 Expansion of 1/sqrt(1 - 4*x/(1+x)^3).

Original entry on oeis.org

1, 2, 0, -4, -4, 6, 18, 4, -48, -70, 60, 288, 170, -686, -1386, 432, 4928, 4806, -9684, -27572, -3672, 84106, 118162, -122388, -537834, -284830, 1386840, 2688944, -1103362, -10181934, -9354198, 21404728, 57921144, 3663942, -185437360, -248708676, 292137656
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2023

Keywords

Links

Crossrefs

Cf. A006139, A137635, A361790, A361791, A361792.

Programs

  • Mathematica
    a[n_]:=(-1)^(n+1)n(n+1)HypergeometricPFQ[{3/2,1-n,1+n/2,(3+n)/2}, {4/3,5/3,2}, 2^4/3^3]; Join[{1},Array[a,36]] (* Stefano Spezia, Jul 11 2024 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^3))

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+2*k-1,n-k).
n*a(n) = -( -2*a(n-1) + (2*n)*a(n-2) + 4*(n-3)*a(n-3) + (n-4)*a(n-4) ) for n > 3.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+1-k,2) * a(k).
a(n) = (-1)^(n+1)*n*(n + 1)*hypergeom([3/2, 1-n, 1+n/2, (3+n)/2], [4/3, 5/3, 2], 2^4/3^3) for n > 0. - Stefano Spezia, Jul 11 2024

A361790 Expansion of 1/sqrt(1 - 4*x/(1+x)^4).

Original entry on oeis.org

1, 2, -2, -8, 6, 42, -8, -228, -90, 1210, 1238, -6116, -10864, 28574, 80932, -116248, -548010, 339678, 3455686, 173208, -20452674, -14036418, 113365140, 156407916, -580805472, -1312098918, 2659610562, 9621079540, -9902139124, -64566648122, 18521111032
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2023

Keywords

Links

Crossrefs

Cf. A006139, A137635, A360133, A361791, A361792.

Programs

  • Mathematica
    a[n_]:=(-1)^(n+1)Pochhammer[n,3]HypergeometricPFQ[{1-n,1+n/3,(4+n)/3, (5+n)/3}, {5/4,7/4,2}, 3^3/2^6]/3; Join[{1},Array[a,30]] (* Stefano Spezia, Jul 11 2024 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^4))
  • PARI
    a(n)=sum(k=0, n, (-1)^(n-k) * binomial(2*k,k) * binomial(n+3*k-1,n-k)) \\ Winston de Greef, Mar 24 2023

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+3*k-1,n-k).
n*a(n) = -( (n-3)*a(n-1) + (6*n-6)*a(n-2) + 10*(n-3)*a(n-3) + 5*(n-4)*a(n-4) + (n-5)*a(n-5) ) for n > 4.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+2-k,3) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,3)*hypergeom([1-n, 1+n/3, (4+n)/3, (5+n)/3], [5/4, 7/4, 2], 3^3/2^6)/3 for n > 0. - Stefano Spezia, Jul 11 2024

A361792 Expansion of 1/sqrt(1 - 4*x/(1+x)^6).

Original entry on oeis.org

1, 2, -6, -10, 66, 60, -750, -236, 8682, -2098, -100792, 80286, 1162458, -1603412, -13225764, 26767020, 147428498, -409582818, -1596563202, 5941802122, 16587101544, -83014131140, -161717252990, 1126247965980, 1411774064970, -14905602076350
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2023

Keywords

Links

Crossrefs

Cf. A006139, A137635, A360133, A361790, A361791.
Cf. A360132.

Programs

  • Mathematica
    a[n_]:=(-1)^(n+1)Pochhammer[n,5]HypergeometricPFQ[{1-n,1+n/5,(6+n)/5, (7+n)/5, (8+n)/5, (9+n)/5}, {7/6,4/3,5/3,11/6,2}, 5^5/(2^4*3^6)]/60; Join[{1},Array[a,25]] (* Stefano Spezia, Jul 11 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^6))

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+5*k-1,n-k).
n*a(n) = -( (3*n-5)*a(n-1) + (17*n-24)*a(n-2) + 35*(n-3)*a(n-3) + 35*(n-4)*a(n-4) + 21*(n-5)*a(n-5) + 7*(n-6)*a(n-6) + (n-7)*a(n-7) ) for n > 6.
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+4-k,5) * a(k).
a(n) = (-1)^(n+1)*Pochhammer(n,5)*hypergeom([1-n, 1+n/5, (6+n)/5, (7+n)/5, (8+n)/5, (9+n)/5], [7/6, 4/3, 5/3, 11/6, 2], 5^5/(2^4*3^6))/60 for n > 0. - Stefano Spezia, Jul 11 2024

A361812 Expansion of 1/sqrt(1 - 4*x*(1+x)^3).

Original entry on oeis.org

1, 2, 12, 62, 342, 1932, 11094, 64480, 378150, 2233304, 13263772, 79136844, 473969586, 2847911596, 17159547804, 103640073972, 627280131594, 3803643145596, 23102172930156, 140522319418164, 855880464524472, 5219168576004184, 31861229045809436
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2023

Keywords

Links

Crossrefs

Column k=3 of A361830.
Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361813, A361814.

Programs

  • Mathematica
    a[n_]:=Binomial[2*n, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 1/2-n, 2/3-n}, -2^6/3^3]; Array[a,23,0] (* Stefano Spezia, Jul 11 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^3))

Formula

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(3*k,n-k).
n*a(n) = 2 * ( (2*n-1)*a(n-1) + 3*(2*n-2)*a(n-2) + 3*(2*n-3)*a(n-3) + (2*n-4)*a(n-4) ) for n > 3.
a(n) = binomial(2*n, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 1/2-n, 2/3-n], -2^6/3^3). - Stefano Spezia, Jul 11 2024

A361813 Expansion of 1/sqrt(1 - 4*x*(1+x)^4).

Original entry on oeis.org

1, 2, 14, 80, 486, 3030, 19184, 122924, 794678, 5173160, 33863666, 222683588, 1469908848, 9733916596, 64636957300, 430240178484, 2869778018070, 19177245746844, 128361805431752, 860443079597872, 5775392952659170, 38811408514848032, 261101034656317244
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2023

Keywords

Crossrefs

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361814.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^4))

Formula

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

A361814 Expansion of 1/sqrt(1 - 4*x*(1+x)^5).

Original entry on oeis.org

1, 2, 16, 100, 660, 4482, 30886, 215364, 1515000, 10730800, 76426846, 546792056, 3926775646, 28290272420, 204375145480, 1479963148220, 10739326203132, 78072933869364, 568503202324540, 4145718464390120, 30271771382355430, 221305746414518180
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2023

Keywords

Links

Crossrefs

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361813.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^5))
  • PARI
    a(n)= sum(k=0, n, binomial(2*k,k) * binomial(5*k,n-k)) \\ Winston de Greef, Mar 25 2023

Formula

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(5*k,n-k).
n*a(n) = 2 * ( (2*n-1)*a(n-1) + 5*(2*n-2)*a(n-2) + 10*(2*n-3)*a(n-3) + 10*(2*n-4)*a(n-4) + 5*(2*n-5)*a(n-5) + (2*n-6)*a(n-6) ) for n > 5.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.