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.

A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)*x)/(1 - 2*(k-1)*x + ((k-3)*x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 2, -15, 1, 4, -6, 32, 1, 6, 9, -12, 105, 1, 8, 30, 16, 30, -576, 1, 10, 57, 140, 25, 60, 105, 1, 12, 90, 384, 630, 36, -140, 5760, 1, 14, 129, 772, 2505, 2772, 49, -280, -13167, 1, 16, 174, 1328, 6430, 16008, 12012, 64, 630, -30400
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2020

Keywords

Examples

			Square array begins:
      1,   1,  1,    1,     1,     1, ...
      0,   2,  4,    6,     8,    10, ...
    -15,  -6,  9,   30,    57,    90, ...
     32, -12, 16,  140,   384,   772, ...
    105,  30, 25,  630,  2505,  6430, ...
   -576,  60, 36, 2772, 16008, 52524, ...
.
From _Peter Luschny_, Jan 20 2020: (Start)
Read by ascending antidiagonals gives:
[0]      1
[1]      0,    1
[2]    -15,    2,  1
[3]     32,   -6,  4,     1
[4]    105,  -12,  9,     6,     1
[5]   -576,   30, 16,    30,     8,    1
[6]    105,   60, 25,   140,    57,   10,    1
[7]   5760, -140, 36,   630,   384,   90,   12,   1
[8] -13167, -280, 49,  2772,  2505,  772,  129,  14,  1
[9] -30400,  630, 64, 12012, 16008, 6430, 1328, 174, 16, 1 (End)

Links

Crossrefs

Columns k=0..5 give A331551, A331552, A000290(n+1), A002457, A108666(n+1), A331323.
T(n,n+3) gives A331512.
Cf. A307883, A331513, A331514.

Programs

  • Maple
    T := (n, k) -> (n + 1)^2*hypergeom([-n, -n], [2], k - 2):
seq(lprint(seq(simplify(T(n,k)), k=0..7)), n=0..6) # Peter Luschny, Jan 20 2020
  • Mathematica
    T[n_, k_] := (n + 1)^2 * HypergeometricPFQ[{-n, -n}, {2}, k - 2];  Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)

Formula

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

A331516 Expansion of 1/(1 - 10*x + 9*x^2)^(3/2).

Original entry on oeis.org

1, 15, 174, 1850, 18855, 187425, 1832460, 17705700, 169569405, 1612842275, 15256106778, 143660483070, 1347716324227, 12603114069525, 117536416879320, 1093553079352200, 10153324144411065, 94098595671581175, 870667876141568070, 8044341506669534850
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Links

Crossrefs

Column 5 of A331514.
Cf. A059231, A084771, A383946.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1 - 10*x + 9*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020
  • Magma
    [1/2*&+[3^(n-k+1)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..20]]; // Marius A. Burtea, Jan 20 2020
  • Mathematica
    a[n_] := 1/2 * Sum[3^( n + 1 - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n+1}]; Array[a, 20, 0] (* Amiram Eldar, Jan 20 2020 *)
CoefficientList[Series[1/(1-10x+9x^2)^(3/2),{x,0,20}],x] (* Harvey P. Dale, Nov 04 2021 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-10*x+9*x^2)^(3/2))
  • PARI
    a(n) = sum(k=1, n+1, 3^(n+1-k)*k*binomial(n+1, k)*binomial(n+1+k, k))/2;

Formula

a(n) = 1/2 * Sum_{k=1..n+1} 3^(n+1-k) * k * binomial(n+1,k) * binomial(n+1+k,k).
n * a(n) = 5 * (2*n+1) * a(n-1) - 9 * (n+1) * a(n-2) for n > 1.
a(n) = ((n+2)/2) * Sum_{k=0..n} 4^k * binomial(n+1,k) * binomial(n+1,k+1).
a(n) ~ sqrt(n) * 3^(2*n + 3) / (2^(7/2) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 9^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 * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A059231(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (5/2)^k * (-9/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)

A331515 Expansion of 1/(1 - 8*x + 4*x^2)^(3/2).

Original entry on oeis.org

1, 12, 114, 1000, 8430, 69384, 561988, 4499856, 35719830, 281634760, 2208564732, 17242680624, 134118558028, 1039939550160, 8041848166920, 62042202765856, 477670318108902, 3670988584476744, 28166853684793420, 215807899372086000, 1651323989374972836
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Links

Crossrefs

Column 4 of A331514.
Cf. A007564, A069835, A385728.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 21); Coefficients(R!( 1/(1 - 8*x + 4*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020
  • Magma
    [&+[2^(n-k)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..21]]; // Marius A. Burtea, Jan 20 2020
  • Mathematica
    a[n_] := Sum[2^(n - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n + 1}]; Array[a, 21, 0] (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(3/2))
  • PARI
    a(n) = sum(k=1, n+1, 2^(n-k)*k*binomial(n+1, k)*binomial(n+1+k, k));

Formula

a(n) = Sum_{k=1..n+1} 2^(n-k) * k * binomial(n+1,k) * binomial(n+1+k,k).
n * a(n) = 4 * (2*n+1) * a(n-1) - 4 * (n+1) * a(n-2) for n>1.
a(n) = ((n+2)/2) * Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).
a(n) ~ 2^(n - 1/2) * (2 + sqrt(3))^(n + 3/2) * sqrt(n) / (3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = binomial(n+2,2) * A007564(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} 2^k * (-1/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)

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)
Showing 1-4 of 4 results.

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