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 1-7 of 7 results.

A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2*k*x + ((k-2)*x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 3, -6, 1, 6, 6, 0, 1, 9, 30, 10, 30, 1, 12, 66, 140, 15, 0, 1, 15, 114, 450, 630, 21, -140, 1, 18, 174, 1000, 2955, 2772, 28, 0, 1, 21, 246, 1850, 8430, 18963, 12012, 36, 630, 1, 24, 330, 3060, 18855, 69384, 119812, 51480, 45, 0
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Examples

			Square array begins:
    1,  1,    1,     1,     1,      1, ...
    0,  3,    6,     9,    12,     15, ...
   -6,  6,   30,    66,   114,    174, ...
    0, 10,  140,   450,  1000,   1850, ...
   30, 15,  630,  2955,  8430,  18855, ...
    0, 21, 2772, 18963, 69384, 187425, ...

Links

Crossrefs

Columns k=1..5 give A000217(n+1), A002457, A002695(n+1), A331515, A331516.
Cf. A307883, A331511.

Programs

  • Mathematica
    T[n_, k_] = 1/2 * Sum[If[k == 2 && n == j - 1, 1, (k - 2)^(n + 1 - j)] * j * Binomial[n + 1, j] * Binomial[n + 1 + j, j], {j, 1, n + 1}]; Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    T(n,k) = (1/2)*sum(j=1,n+1,(k-2)^(n+1-j)*j*binomial(n+1,j)*binomial(n+1+j,j));
matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jan 20 2020

Formula

T(n,k) = (1/2) * Sum_{j=1..n+1} (k-2)^(n+1-j) * j * binomial(n+1,j) * binomial(n+1+j,j).
n * T(n,k) = k * (2*n+1) * T(n-1,k) - (k-2)^2 * (n+1) * T(n-2,k) for n > 1.
T(n,k) = ((n+2)/2) * Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
T(n,k) = Sum_{j=0..n} (k/2)^j * (-(k-2)^2/(2*k))^(n-j) * (2*j+1) * binomial(2*j,j) * binomial(j,n-j) for k > 0. - Seiichi Manyama, Aug 20 2025

A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2*k*x + ((k-2)*x)^2 + (1 - k*x) * sqrt(1 - 2*k*x + ((k-2)*x)^2)).

Original entry on oeis.org

1, 1, 0, 1, 2, -3, 1, 4, 3, 0, 1, 6, 15, 4, 10, 1, 8, 33, 56, 5, 0, 1, 10, 57, 180, 210, 6, -35, 1, 12, 87, 400, 985, 792, 7, 0, 1, 14, 123, 740, 2810, 5418, 3003, 8, 126, 1, 16, 165, 1224, 6285, 19824, 29953, 11440, 9, 0, 1, 18, 213, 1876, 12130, 53550, 140497, 166344, 43758, 10, -462
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2020

Keywords

Examples

			Square array begins:
   1, 1,   1,    1,     1,     1, ...
   0, 2,   4,    6,     8,    10, ...
  -3, 3,  15,   33,    57,    87, ...
   0, 4,  56,  180,   400,   740, ...
  10, 5, 210,  985,  2810,  6285, ...
   0, 6, 792, 5418, 19824, 53550, ...

Links

Crossrefs

Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.
T(n,n+1) gives A331794.
Cf. A307883, A331511, A331514, A331795.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k==1 && j==0, 1, (k-1)^j] * Binomial[n + 1, j] * Binomial[n + 1, j + 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 2021 *)

Formula

T(n,k) = Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
n * (n+2) * T(n,k) = (n+1) * (k * (2*n+1) * T(n-1,k) - (k-2)^2 * n * T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (k-1)^j * k^(n-2*j) * binomial(n+1,n-2*j) * binomial(2*j+1,j). - Seiichi Manyama, Aug 24 2025
From Seiichi Manyama, Aug 27 2025: (Start)
T(n,k) = [x^n] (1+k*x+(k-1)*x^2)^(n+1).
For k != 1, e.g.f. of column k: exp(k*x) * BesselI(1, 2*sqrt(k-1)*x) / sqrt(k-1), with offset 1. (End)

A331323 a(n) = [x^n] (1 - 2*x)/(1 - 8*x + 4*x^2)^(3/2).

Original entry on oeis.org

1, 10, 90, 772, 6430, 52524, 423220, 3375880, 26720118, 210195100, 1645295212, 12825551160, 99633196780, 771702434104, 5961969066600, 45958506432016, 353585912577190, 2715647948258940, 20824876515839932, 159474192002499160, 1219708190630800836, 9318143974952519080
Offset: 0

Views

Author

Peter Luschny, Jan 18 2020

Keywords

Links

Crossrefs

Column 5 of A331511.
Cf. A331431.

Programs

  • Magma
    [(&+[3^k*(k+1)*Binomial(n+1,k+1)^2: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Mar 22 2022
  • Maple
    gf := (1-2*x)/(4*x^2-8*x+1)^(3/2): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=0..21); # Or:
a := proc(n) option remember; if n<3 then [1, 10, 90][n+1] else
(10*n*a(n-1) + 20*(1-n)*a(n-2) + 8*(n-1)*a(n-3))/n fi end:
seq(a(n), n=0..21);
  • Mathematica
    a[n_] := Sum[3^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    {a(n) = 2^n*sum(k=0, n, (n+k+1)*binomial(n, k)*binomial(n+k, k)/2^k)} \\ Seiichi Manyama, Jan 18 2020
  • PARI
    N=20; x='x+O('x^N); Vec((1-2*x)/(4*x^2-8*x+1)^(3/2)) \\ Seiichi Manyama, Jan 18 2020
  • PARI
    {a(n) = sum(k=0, n, 3^k*(k+1)*binomial(n+1, k+1)^2)} \\ Seiichi Manyama, Jan 20 2020
  • Sage
    [sum(2^(n-k)*(n+k+1)*binomial(2*k,k)*binomial(n+k,2*k)  for k in (0..n)) for n in (0..30)] # G. C. Greubel, Mar 22 2022

Formula

a(n) = (-2)^n*Sum_{k=0..n} A331431(n, k)/(-2)^k.
a(n) = (10*n*a(n-1) + 20*(1-n)*a(n-2) + 8*(n-1)*a(n-3))/n.
a(n) = 2^(n-1)*(n+1)*(n*hypergeom([1-n, n+2], [2], -1/2) + 2*hypergeom([-n, n+1], [1], -1/2)).
a(n) = Sum_{k=0..n} 3^k * (k+1) * binomial(n+1,k+1)^2. - Seiichi Manyama, Jan 20 2020
a(n) = (n + 1)^2*hypergeom([-n, -n], [2], 3). - Peter Luschny, Jan 20 2020
n * (2*n-1) * a(n) = 2 * (8 * n^2 - 3) * a(n-1) - 4 * n * (2*n+1) * a(n-2) for n>1. - Seiichi Manyama, Jan 25 2020

A331512 a(n) = Sum_{k=0..n} n^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).

Original entry on oeis.org

1, 8, 90, 1328, 24150, 520272, 12926004, 363233600, 11376760230, 392615960600, 14791582824876, 603743206301424, 26528443526357500, 1248071683342913184, 62576263671773466600, 3330116426356595493120, 187430800395881065513734
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Links

Crossrefs

Cf. A001850, A108666, A331511, A331513.

Programs

  • Mathematica
    a[n_] := Sum[If[n == n-k == 0, 1, n^(n-k)] * (n+k+1) * Binomial[n, k] * Binomial[n + k, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, May 05 2021 *)
  • PARI
    {a(n) = sum(k=0, n, n^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
  • PARI
    {a(n) = polcoef((1-n*x)/(1-2*(n+2)*x+(n*x)^2)^(3/2), n)}
  • PARI
    {a(n) = sum(k=0, n, (n+1)^k*(k+1)*binomial(n+1, k+1)^2)}

Formula

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

A331513 a(n) = Sum_{k=0..n} (-n)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).

Original entry on oeis.org

1, 4, -6, 32, -170, -228, 43764, -1498880, 43826598, -1249865260, 35978752876, -1053020066976, 31153402105852, -914722450924436, 25562930671296360, -604802562457466880, 5868775340572918534, 684246820455046681380, -78372285809430441261828
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Links

Crossrefs

Cf. A331511, A331512.

Programs

  • Mathematica
    a[n_] := Sum[If[n == n-k == 0, 1, (-n)^(n-k)] * (n+k+1) * Binomial[n, k] * Binomial[n + k, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, May 05 2021 *)
  • PARI
    {a(n) = sum(k=0, n, (-n)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
  • PARI
    {a(n) = polcoef((1+n*x)/(1+2*(n-2)*x+(n*x)^2)^(3/2), n)}
  • PARI
    {a(n) = sum(k=0, n, (-n+1)^k*(k+1)*binomial(n+1, k+1)^2)}

Formula

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

A331551 Expansion of (1 + 3*x)/(1 + 2*x + 9*x^2)^(3/2).

Original entry on oeis.org

1, 0, -15, 32, 105, -576, 105, 5760, -13167, -30400, 194337, -104160, -1685255, 4497024, 7011225, -57705984, 51497505, 445080960, -1402731183, -1348950240, 16032154761, -20039110080, -110074987575, 412984420992, 190753103025
Offset: 0

Views

Author

Seiichi Manyama, Jan 20 2020

Keywords

Links

Crossrefs

Column 0 of A331511.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 + 3*x)/(1 + 2*x + 9*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020
  • Maple
    a := n -> (n + 1)^2*hypergeom([-n, -n], [2], -2):
seq(simplify(a(n)), n=0..19); # Peter Luschny, Jan 20 2020
  • Mathematica
    a[n_] := Sum[(-2)^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    N=20; x='x+O('x^N); Vec((1+3*x)/(1+2*x+9*x^2)^(3/2))
  • PARI
    {a(n) = sum(k=0, n, (-3)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
  • PARI
    {a(n) = sum(k=0, n, (-2)^k*(k+1)*binomial(n+1, k+1)^2)}

Formula

a(n) = Sum_{k=0..n} (-3)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).
a(n) = Sum_{k=0..n} (-2)^k * (k+1) * binomial(n+1,k+1)^2.
a(n) = (n + 1)^2*hypergeom([-n, -n], [2], -2). - Peter Luschny, Jan 20 2020
n * (2*n-1) * a(n) = 4 * (-n^2 + 1) * a(n-1) - 9 * n * (2*n+1) * a(n-2) for n>1. - Seiichi Manyama, Jan 25 2020

A331552 Expansion of (1 + 2*x)/(1 + 4*x^2)^(3/2).

Original entry on oeis.org

1, 2, -6, -12, 30, 60, -140, -280, 630, 1260, -2772, -5544, 12012, 24024, -51480, -102960, 218790, 437580, -923780, -1847560, 3879876, 7759752, -16224936, -32449872, 67603900, 135207800, -280816200, -561632400, 1163381400, 2326762800, -4808643120, -9617286240, 19835652870
Offset: 0

Views

Author

Seiichi Manyama, Jan 20 2020

Keywords

Links

Crossrefs

Column 1 of A331511.
Cf. A100071.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 33); Coefficients(R!( (1 + 2*x)/(1 + 4*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020
  • Magma
    [&+[(-1)^k*(k+1)*Binomial(n+1, k+1)^2:k in [0..n]]:n in [0..33]]; // Marius A. Burtea, Jan 20 2020
  • Mathematica
    a[n_] := Sum[(-1)^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 33, 0] (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    N=66; x='x+O('x^N); Vec((1+2*x)/(1+4*x^2)^(3/2))
  • PARI
    {a(n) = sum(k=0, n, (-2)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
  • PARI
    {a(n) = sum(k=0, n, (-1)^k*(k+1)*binomial(n+1, k+1)^2)}

Formula

|a(n)| = A100071(n+1).
a(n) = Sum_{k=0..n} (-2)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).
a(n) = Sum_{k=0..n} (-1)^k * (k+1) * binomial(n+1,k+1)^2.
n * (2*n-1) * a(n) = 2 * a(n-1) - 4 * n * (2*n+1) * a(n-2) for n>1.
E.g.f.: (1 + 2*x)*BesselJ(0,2*x) - 2*x*BesselJ(1,2*x). - Ilya Gutkovskiy, Mar 04 2021
Showing 1-7 of 7 results.

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