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.

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A308500 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=0..n} binomial(k*n,k*j).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 8, 8, 5, 1, 2, 22, 32, 16, 6, 1, 2, 72, 170, 128, 32, 7, 1, 2, 254, 992, 1366, 512, 64, 8, 1, 2, 926, 6008, 16512, 10922, 2048, 128, 9, 1, 2, 3434, 37130, 215766, 261632, 87382, 8192, 256, 10, 1, 2, 12872, 232562, 2973350, 6643782, 4196352, 699050, 32768, 512, 11
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2019

Keywords

Examples

			Square array begins:
   1,  1,    1,     1,       1,         1, ...
   2,  2,    2,     2,       2,         2, ...
   3,  4,    8,    22,      72,       254, ...
   4,  8,   32,   170,     992,      6008, ...
   5, 16,  128,  1366,   16512,    215766, ...
   6, 32,  512, 10922,  261632,   6643782, ...
   7, 64, 2048, 87382, 4196352, 215492564, ...

Links

Crossrefs

Columns k=0..10 give A000027(n+1), A000079, A081294, A007613, A070775, A070782, A070967, A094211, A070832, A094213, A070833.
Main diagonal gives A167009.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[k*n, k*j], {j, 0, n}] ; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A348315 a(n) = Sum_{k=0..n} binomial(n^2 - k,n*k).

Original entry on oeis.org

1, 1, 4, 64, 4382, 1357136, 1597653852, 8389021518585, 164828345435877580, 14256525628649472111712, 4602970880920727147946847283, 6484132480933772335644792339409450, 34112054985056318746734374876035089268523
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2021

Keywords

Links

Crossrefs

Cf. A167009, A238696, A306680, A348322.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n^2 - k, n*k], {k, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, Oct 12 2021 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n^2-k, n*k));
  • PARI
    a(n) = polcoef((1-x)^(n-1)/((1-x)^n-x^(n+1)+x*O(x^n^2)), n^2);

Formula

a(n) = A306680(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^(n+1)).
a(n) ~ c * 2^(1/2 - n/2 + n^2) / (sqrt(Pi)*exp(1/8)*n), where c = Sum_{m = -oo..+oo} 1/(2^m * exp(m*(2*m+1))) = 1.77058122254033174512511... if n is even and c = Sum_{m = -oo..+oo} 1/(2^(m + 1/2) * exp((m+1)*(2*m+1))) = 1.81629595919505881855931... if n is odd. - Vaclav Kotesovec, Oct 12 2021

A207140 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2).

Original entry on oeis.org

1, 2, 10, 407, 56746, 30771252, 115106662819, 1446405270234360, 53819202633553797290, 12313337704248075967333334, 12373818231445938048765251252260, 33156027144321617106970597265032233270, 409476940913917468665022448013012674533441891
Offset: 0

Views

Author

Paul D. Hanna, Feb 15 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A207139.

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 407*x^3/3 + 56746*x^4/4 + 30771252*x^5/5 +...
where exponentiation equals the g.f. of A207139:
exp(L(x)) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
By definition, the initial terms begin: a(0) = 1;
a(1) = C(1,0)*C(1,0), + C(1,1)*C(1,1);
a(2) = C(2,0)*C(4,0), + C(2,1)*C(4,1), + C(2,2)*C(4,4);
a(3) = C(3,0)*C(9,0), + C(3,1)*C(9,1), + C(3,2)*C(9,4), + C(3,3)*C(9,9);
a(4) = C(4,0)*C(16,0), + C(4,1)*C(16,1), + C(4,2)*C(16,4), + C(4,3)*C(16,9), + C(4,4)*C(16,16); ...
which is evaluated as:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 2*4 + 1*1 = 10;
a(3) = 1*1 + 3*9 + 3*126 + 1*1 = 407;
a(4) = 1*1 + 4*16 + 6*1820 + 4*11440 + 1*1 = 56746;
a(5) = 1*1 + 5*25 + 10*12650 + 10*2042975 + 5*2042975 + 1*1 = 30771252;
a(6) = 1*1 + 6*36 + 15*58905 + 20*94143280 + 15*7307872110 + 6*600805296 + 1*1 = 115106662819; ...

Crossrefs

Cf. A207139 (exp), A206851, A207136, A207138, A167009.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * Binomial[n^2,k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k^2))}
for(n=0,16,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = 2. - Vaclav Kotesovec, Mar 03 2014
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