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 1-10 of 11 results. Next

A336573 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (-1)^n * Sum_{j=0..n} (-2)^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 8, 1, 1, 5, 21, 45, 16, 1, 1, 6, 34, 126, 197, 32, 1, 1, 7, 50, 267, 818, 903, 64, 1, 1, 8, 69, 484, 2279, 5594, 4279, 128, 1, 1, 9, 91, 793, 5105, 20540, 39693, 20793, 256, 1, 1, 10, 116, 1210, 9946, 56928, 192350, 289510, 103049, 512
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2020

Keywords

Comments

T(n,k) is the number of Sylvester classes of k-packed words of degree n.

Examples

			Square array begins:
   1,   1,   1,    1,    1,    1, ...
   1,   1,   1,    1,    1,    1, ...
   2,   3,   4,    5,    6,    7, ...
   4,  11,  21,   34,   50,   69, ...
   8,  45, 126,  267,  484,  793, ...
  16, 197, 818, 2279, 5105, 9946, ...

Links

Crossrefs

Columns k = 0-5 are: A011782, A001003, A003168, A243659, A243667, A243668.
Main diagonal is A336495.
Cf. A336534, A336574, A336575.

Programs

  • Maple
    T := (n,k) -> `if`(k=0, `if`(n=0, 1, 2^(n-1)), (-1)^n*(binomial(k*n+1, n)* hypergeom([-n, k*n+1], [(k-1)*n+2], 2)) / (k*n+1)):
seq(lprint(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, Jul 26 2020
  • Mathematica
    T[n_, k_] := (-1)^n * Sum[(-2)^j * Binomial[n, j] * Binomial[k*n+j+1, n]/(k*n+j+1), {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
  • PARI
    T(n, k) = (-1)^n*sum(j=0, n, (-2)^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));
  • PARI
    T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^k*(1-2*A)); polcoeff(A, n);
  • PARI
    T(n, k) = (-1)^n*sum(j=0, n, (-2)^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 - x * A_k(x)^k * (1 - 2 * A_k(x)).
T(n,k) = ( (-1)^n / (k*n+1) ) * Sum_{j=0..n} (-2)^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
T(n,k) = (-1)^n*binomial(k*n+1, n)*hypergeom([-n, k*n+1], [(k-1)*n+2], 2)/(k*n+1) for k >= 1. - Peter Luschny, Jul 26 2020
T(n,k) = (1/n) * Sum_{j=0..n-1} binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0. - Seiichi Manyama, Aug 08 2023

A336574 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

Original entry on oeis.org

1, 1, 3, 1, 3, 6, 1, 3, 15, 12, 1, 3, 24, 93, 24, 1, 3, 33, 255, 645, 48, 1, 3, 42, 498, 3102, 4791, 96, 1, 3, 51, 822, 8691, 40854, 37275, 192, 1, 3, 60, 1227, 18708, 164937, 566934, 299865, 384, 1, 3, 69, 1713, 34449, 464115, 3305868, 8164263, 2474025, 768
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2020

Keywords

Examples

			Square array begins:
   1,    1,     1,      1,      1,       1, ...
   3,    3,     3,      3,      3,       3, ...
   6,   15,    24,     33,     42,      51, ...
  12,   93,   255,    498,    822,    1227, ...
  24,  645,  3102,   8691,  18708,   34449, ...
  48, 4791, 40854, 164937, 464115, 1055838, ...

Links

Crossrefs

Columns k=0-4 give: A003945, A103210, A219536, A336539, A336572.
Main diagonal gives A336577.
Cf. A336534, A336573, A336575.

Programs

  • Mathematica
    T[n_, k_] := Sum[2^j * Binomial[n, j] * Binomial[k*n + j + 1, n]/(k*n + j + 1), {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 27 2020 *)
  • PARI
    T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));
  • PARI
    T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k*(1+2*A)); polcoeff(A, n);
  • PARI
    T(n, k) = sum(j=0, n, 2^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (1 + 2 * A_k(x)).
T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} 2^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
From Seiichi Manyama, Aug 10 2023: (Start)
T(n,k) = (1/n) * Sum_{j=0..n-1} (-1)^j * 3^(n-j) * binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0.
T(n,k) = (1/n) * Sum_{j=1..n} 3^j * 2^(n-j) * binomial(n,j) * binomial(k*n,j-1) for n > 0. (End)
T(n,k) = binomial(1+k*n, n)*hypergeom([-n, 1+k*n], [2+(k-1)*n], -2)/(1 + k*n) for k > 0. - Stefano Spezia, Aug 09 2025

A336578 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(n^2,k-1) for n > 0.

Original entry on oeis.org

1, 3, 21, 408, 14799, 817743, 61621806, 5921141502, 694008501627, 96176405390961, 15400332946269903, 2799678523675400832, 569877183695866859625, 128436925725088289658534, 31756620986815666396814796, 8548059658831271609064999978, 2488568825786280454788465874035, 779186768737628124697943895022101
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2020

Keywords

Links

Crossrefs

Main diagonal of A336575.
Cf. A336495, A336537, A336577.

Programs

  • Mathematica
    a[0] := 1; a[n_] := Sum[3^k * Binomial[n, k] * Binomial[n^2, k - 1], {k, 1, n}]/n;  Array[a, 18, 0] (* Amiram Eldar, Jul 27 2020 *)
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, 3^k*binomial(n, k)*binomial(n^2, k-1))/n);
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(n^2+k+1, n)/(n^2+k+1)); \\ Seiichi Manyama, Jul 27 2020
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(n^2+1, k)*binomial((n+1)*n-k, n-k))/(n^2+1); \\ Seiichi Manyama, Jul 27 2020

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).
a(n) = (1/(n^2+1)) * Sum_{k=0..n} 2^k * binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).
a(n) ~ 3^n * exp(n - 1/6) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Jul 31 2021
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial((n+1)*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 10 2023
a(n) = 3*hypergeom([1-n, -n^2], [2], 3) for n > 0. - Stefano Spezia, Aug 09 2025

A219535 G.f. satisfies A(x) = 1 + x*(2*A(x)^2 + A(x)^3).

Original entry on oeis.org

1, 3, 21, 192, 2001, 22539, 267276, 3287496, 41556585, 536565225, 7046232285, 93820316412, 1263673602300, 17186898452772, 235709926636296, 3256050894487824, 45263067114496665, 632721425905230213, 8888476706476318047, 125418490224196533096, 1776734673565844413929
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2012

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 192*x^3 + 2001*x^4 + 22539*x^5 +...
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 510*x^3 + 5595*x^4 + 65148*x^5 +...
A(x)^3 = 1 + 9*x + 90*x^2 + 981*x^3 + 11349*x^4 + 136980*x^5 +...
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + 3*x + 12*x^2 + 57*x^3 + 300*x^4 + 1686*x^5 +...+ A047891(n+1)*x^n +...

Links

Crossrefs

Column k=2 of A336575.
Cf. A047891, A219534, A219536.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[2*x^2/(1-2*x-Sqrt[1-8*x+4*x^2]), {x, 0, 21}], x],x] (* Vaclav Kotesovec, Dec 28 2013 *)
  • PARI
    /* Formula A(x) = 1 + x*(2*A(x)^2 + A(x)^3): */
{a(n)=my(A=1);for(i=1,n,A=1+x*(2*A^2+A^3) +x*O(x^n));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula using Series Reversion: */
{a(n)=my(A=1,G=(1-2*x-sqrt(1-8*x+4*x^2+x^3*O(x^n)))/(2*x));A=(1/x)*serreverse(x/G);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ Seiichi Manyama, Jul 28 2020
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(2*n+1, k)*binomial(3*n-k, n-k))/(2*n+1); \\ Seiichi Manyama, Jul 28 2020

Formula

Let G(x) = (1-2*x - sqrt(1 - 8*x + 4*x^2)) / (2*x), then g.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x/G(x)),
(2) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
where x*G(x) is the g.f. of A047891.
Recurrence: 2*n*(2*n+1)*(11*n - 16)*a(n) = (649*n^3 - 1593*n^2 + 1130*n - 240)*a(n-1) + 16*(n-2)*(2*n-3)*(11*n-5)*a(n-2). - Vaclav Kotesovec, Dec 28 2013
a(n) ~ sqrt((33+17*sqrt(33))/11) * ((59+11*sqrt(33))/8)^n / (4 * sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 28 2013
From Seiichi Manyama, Jul 28 2020: (Start)
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(2*n+k+1,n)/(2*n+k+1).
a(n) = (1/(2*n+1)) * Sum_{k=0..n} 2^k * binomial(2*n+1,k) * binomial(3*n-k,n-k). (End)
From Seiichi Manyama, Aug 10 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)

A336534 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 6, 2, 1, 2, 10, 22, 2, 1, 2, 14, 66, 90, 2, 1, 2, 18, 134, 498, 394, 2, 1, 2, 22, 226, 1482, 4066, 1806, 2, 1, 2, 26, 342, 3298, 17818, 34970, 8558, 2, 1, 2, 30, 482, 6202, 52450, 226214, 312066, 41586, 2, 1, 2, 34, 646, 10450, 122762, 881970, 2984206, 2862562, 206098, 2
Offset: 0

Views

Author

Seiichi Manyama, Jul 25 2020

Keywords

Examples

			Square array begins:
  1,   1,    1,     1,     1,      1, ...
  2,   2,    2,     2,     2,      2, ...
  2,   6,   10,    14,    18,     22, ...
  2,  22,   66,   134,   226,    342, ...
  2,  90,  498,  1482,  3298,   6202, ...
  2, 394, 4066, 17818, 52450, 122762, ...

Links

Crossrefs

Columns k=0-3 give A040000, A006318, A027307, A144097.
If Michael D. Weiner's conjecture on A260332 is correct, column 4 is A260332 for n > 0.
Main diagonal gives A336537.
Cf. A336521, A336574. A336575.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n+j+1, n]/(k*n+j+1), {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
  • PARI
    T(n, k) = sum(j=0, n, binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (1 + A_k(x)).
T(n,k) = (1/n) * Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n,j-1) for n > 0.
T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
T(n,k) = binomial(1+k*n, n)*hypergeom([-n, 1+k*n], [2+(k-1)*n], -1)/(1 + k*n) for k > 0. - Stefano Spezia, Aug 09 2025

A336708 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-1)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, 1, 1, -1, 2, 1, 1, 2, 1, 0, -3, 1, 1, 3, 6, 1, 2, -1, 1, 1, 4, 14, 21, 1, 0, 11, 1, 1, 5, 25, 76, 80, 1, -5, -15, 1, 1, 6, 39, 182, 450, 322, 1, 0, -13, 1, 1, 7, 56, 355, 1447, 2818, 1347, 1, 14, 77, 1, 1, 8, 76, 611, 3532, 12175, 18352, 5798, 1, 0, -86
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2020

Keywords

Examples

			Square array begins:
   1,  1, 1,   1,    1,     1,     1, ...
   1,  1, 1,   1,    1,     1,     1, ...
  -1,  0, 1,   2,    3,     4,     5, ...
   0, -1, 1,   6,   14,    25,    39, ...
   2,  0, 1,  21,   76,   182,   355, ...
  -3,  2, 1,  80,  450,  1447,  3532, ...
  -1,  0, 1, 322, 2818, 12175, 37206, ...

Links

Crossrefs

Columns k=0-3 give: A007440, A090192, A000012, A106228.
Main diagonal gives A336713.
Cf. A336575, A336706, A336707, A336709.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    {T(n, k) = if(n==0, 1, sum(j=1, n, (-1)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
  • PARI
    {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+x*A)); polcoef(A, n)}

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + x * A_k(x)).

Extensions

Typo in name corrected by Georg Fischer, Sep 19 2023

A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, -2, 1, 1, -1, 2, 1, 1, 0, -1, 4, 1, 1, 1, -1, 5, -24, 1, 1, 2, 2, 0, -3, 48, 1, 1, 3, 8, 5, 2, -21, 24, 1, 1, 4, 17, 36, 13, 0, 51, -464, 1, 1, 5, 29, 109, 177, 36, -5, 41, 1376, 1, 1, 6, 44, 240, 766, 922, 104, 0, -391, -704
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2020

Keywords

Examples

			Square array begins:
    1,   1,  1,  1,   1,    1,     1, ...
    1,   1,  1,  1,   1,    1,     1, ...
   -2,  -1,  0,  1,   2,    3,     4, ...
    2,  -1, -1,  2,   8,   17,    29, ...
    4,   5,  0,  5,  36,  109,   240, ...
  -24,  -3,  2, 13, 177,  766,  2177, ...
   48, -21,  0, 36, 922, 5699, 20910, ...

Links

Crossrefs

Columns k=0-3 give: A307969(n-1), (-1)^n * A154825(n), A090192, A246555.
Main diagonal gives A336714.
Cf. A336575, A336706, A336707, A336708.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[(-2)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    {T(n, k) = if(n==0, 1, sum(j=1, n, (-2)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
  • PARI
    {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+2*x*A)); polcoef(A, n)}

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + 2 * x * A_k(x)).

A336706 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 4, 1, 1, 4, 11, 14, 9, 1, 1, 5, 20, 45, 42, 21, 1, 1, 6, 32, 113, 197, 132, 51, 1, 1, 7, 47, 234, 688, 903, 429, 127, 1, 1, 8, 65, 424, 1854, 4404, 4279, 1430, 323, 1, 1, 9, 86, 699, 4159, 15490, 29219, 20793, 4862, 835
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2020

Keywords

Examples

			Square array begins:
   1,   1,   1,    1,     1,     1,      1, ...
   1,   1,   1,    1,     1,     1,      1, ...
   1,   2,   3,    4,     5,     6,      7, ...
   2,   5,  11,   20,    32,    47,     65, ...
   4,  14,  45,  113,   234,   424,    699, ...
   9,  42, 197,  688,  1854,  4159,   8192, ...
  21, 132, 903, 4404, 15490, 43097, 101538, ...

Links

Crossrefs

Columns k=0-3 give: A001006(n-1), A000108, A001003, A108447.
Main diagonal gives A335871.
Cf. A336575, A336707, A336708, A336709.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    {T(n, k) = if(n==0, 1, sum(j=1, n, binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
  • PARI
    {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1-x*A)); polcoef(A, n)}

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 - x * A_k(x)).

A336707 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} 2^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 11, 20, 1, 1, 5, 19, 45, 72, 1, 1, 6, 30, 100, 197, 272, 1, 1, 7, 44, 201, 562, 903, 1064, 1, 1, 8, 61, 364, 1445, 3304, 4279, 4272, 1, 1, 9, 81, 605, 3249, 10900, 20071, 20793, 17504, 1, 1, 10, 104, 940, 6502, 30526, 85128, 124996, 103049, 72896
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2020

Keywords

Examples

			Square array begins:
    1,   1,    1,     1,     1,     1,      1, ...
    1,   1,    1,     1,     1,     1,      1, ...
    2,   3,    4,     5,     6,     7,      8, ...
    6,  11,   19,    30,    44,    61,     81, ...
   20,  45,  100,   201,   364,   605,    940, ...
   72, 197,  562,  1445,  3249,  6502,  11857, ...
  272, 903, 3304, 10900, 30526, 73723, 158034, ...

Links

Crossrefs

Columns k=0-3 give: A071356(n-1), A001003, A007564, A118346.
Main diagonal gives A336712.
Cf. A336575, A336706, A336708, A336709.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[2^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    {T(n, k) = if(n==0, 1, sum(j=1, n, 2^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
  • PARI
    {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1-2*x*A)); polcoef(A, n)}

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 - 2 * x * A_k(x)).

A336538 G.f. A(x) satisfies A(x) = 1 + x * A(x)^3 * (2 + A(x)).

Original entry on oeis.org

1, 3, 30, 408, 6402, 109137, 1964010, 36718680, 706221210, 13883562732, 277730910840, 5635185129696, 115693119210270, 2398955889524934, 50167967688522012, 1056869531313301200, 22407983968252808586, 477791976566108489700, 10238908702033904618856, 220401923906465000263200, 4763512100782704414532296
Offset: 0

Views

Author

Seiichi Manyama, Jul 25 2020

Keywords

Links

Crossrefs

Column k=3 of A336575.

Programs

  • Mathematica
    a[n_] := Sum[2^(n-k) * Binomial[n, k] * Binomial[3*n + k + 1, n]/(3*n + k + 1), {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Jul 28 2020 *)
  • PARI
    a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^3*(2+A)); polcoeff(A, n);
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, 3^k*binomial(n, k)*binomial(3*n, k-1)/n));
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1)); \\ Seiichi Manyama, Jul 28 2020
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(3*n+1, k)*binomial(4*n-k, n-k))/(3*n+1); \\ Seiichi Manyama, Jul 28 2020

Formula

a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(3*n,k-1) for n > 0.
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1).
a(n) = (1/(3*n+1)) * Sum_{k=0..n} 2^k * binomial(3*n+1,k) * binomial(4*n-k,n-k).
a(n) ~ (12 + 8*sqrt(2))^n / (2^(3/4) * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 31 2021
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 10 2023
a(n) = 2^n*binomial(1+3*n, n)*hypergeom([-n, 1+3*n], [2*(1+n)], -1/2)/(1 + 3*n). - Stefano Spezia, Aug 09 2025
Showing 1-10 of 11 results. Next

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

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