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 1-3 of 3 results.

A134772 Let f(m, n) = 2^(-m) * Sum_{j=0..n} (-1)^j*m!*n!*(2*m-2*j)!/(j!*(m-j)! *(n-j)!*6^(n-j)) then a(n) = f(2n,n).

Original entry on oeis.org

1, 0, 28, 14400, 27165600, 134289792000, 1445549490000000, 29865588836219136000, 1081114481157129619200000, 64007711015400701105356800000, 5873237165016878140678626432000000, 799901135455942846519287494400000000000, 156064894765355001368149078831725782016000000
Offset: 0

Views

Author

N. J. A. Sloane, Oct 18 2009

Keywords

Links

Crossrefs

A variant of A132202.
Bisections: A137942, A144649.
Cf. A134648, A152296.

Programs

  • Magma
    B:=Binomial;
A134772:= func< n | Factorial(4*n)/(24)^n *(&+[B(n,j)*B(2*n,j)*(-6)^j/(Factorial(j)*B(2*j,j)*B(4*n,2*j)) : j in [0..n]]) >;
[A134772(n): n in [0..30]]; // G. C. Greubel, Oct 12 2023
  • Mathematica
    Table[((4*n)!/(24)^n)*Hypergeometric1F1[-n, 1/2-2*n, -3/2], {n,0,30}] (* G. C. Greubel, Oct 12 2023 *)
  • SageMath
    def A134772(n): return (factorial(4*n)/(24)^n)* simplify(hypergeometric([-n], [1/2-2*n], -3/2))
[A134772(n) for n in range(31)] # G. C. Greubel, Oct 12 2023

Formula

a(n) = 4^(-n) * Sum_{j=0..n} (-1)^j*(2*n)!*n!*(4*n-2*j)!/(j!*(2*n-j)! *(n-j)!*6^(n-j)).
From G. C. Greubel, Oct 12 2023: (Start)
a(n) = ((4*n)!/(24)^n) * Sum_{j=0..n} b(n,j)*b(2*n,j)(-6)^j/(b(2*j,j) * b(4*n,2*j)), where b(x,y) = binomial(x,y).
a(n) = ((4*n)!/(24)^n) * Hypergeometric1F1([-n], [1/2 -2*n], -3/2).
Sum_{n>=0} a(n)*x^n/(n!*(2*n)!) = 1/sqrt(1+x) * Hypergeometric2F0([1/4, 3/4]; --; 8*x/(3*(1+x)^2)). (End)
a(n) ~ sqrt(Pi) * 2^(5*n + 3/2) * n^(4*n + 1/2) / (3^n * exp(4*n + 3/4)). - Vaclav Kotesovec, Oct 21 2023

A132202 Number of 3n X 2n (0,1)-matrices with every row sum 2 and column sum 3.

Original entry on oeis.org

1, 1860, 90291600, 31082452632000, 46764764308702440000, 229747284991066934931840000, 3031982831164890119435183865600000, 93453554057243260025029337978773248000000, 6055976192395031960092036887782708145734400000000, 760152286561053082358524425840024164536832608896000000000
Offset: 1

Views

Author

Shanzhen Gao, Nov 05 2007

Keywords

Examples

			1 for 3X2:
  11
  11
  11
1860 for 6X4.
90291600 for 9X6.

References

  • Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.

Links

Crossrefs

Cf. A134648, A134772.

Programs

  • Magma
    B:=Binomial;
A132202:= func< n | Factorial(6*n)/(288)^n*(&+[B(2*n,j)*B(3*n,j)*(-6)^j/(Factorial(j)*B(2*j,j)*B(6*n,2*j)): j in [0..2*n]]) >;
[A132202(n): n in [1..30]]; // G. C. Greubel, Oct 12 2023
  • Maple
    f:=proc(m,n) 2^(-m)*add( ((-1)^(i)*m!*n!*(2*m-2*i)!)/ (i!*(m-i)!*(n-i)!*6^(n-i)), i=0..n); end;
[seq(f(3*n,2*n),n=0..10)];
  • Mathematica
    Table[((6*n)!/(288)^n)*Hypergeometric1F1[-2*n,1/2-3*n,-3/2], {n,30}] (* G. C. Greubel, Oct 12 2023 *)
  • SageMath
    b=binomial
def A132202(n): return factorial(6*n)/(288)^n *simplify(hypergeometric([-2*n], [1/2-3*n], -3/2))
[A132202(n) for n in range(1,31)] # G. C. Greubel, Oct 12 2023

Formula

a(n) = f(3*n, 2*n), where f(m, n) = 2^(-m) * Sum_{j=0..n} (-1)^j*n!*m!*(2*m-2*j)!/(j!*(m-j)!*(n-j)!*6^(n-j)).
From G. C. Greubel, Oct 12 2023: (Start)
a(n) = ((6*n)!/(288)^n)*Sum_{j=0..2*n} b(2*n,j)*b(3*n,j)*(-6)^j/(j!*b(2*j, j)*b(6*n,2*j)), where b(x,y) = binomomial(x,y).
a(n) = (6*n)!/(288)^n * Hypergeometric1F1([-2*n], [1/2-3*n], -3/2). (End)
a(n) ~ sqrt(Pi) * 2^(n+1) * 3^(4*n + 1/2) * n^(6*n + 1/2) / exp(6*n+1). - Vaclav Kotesovec, Oct 21 2023

Extensions

Edited and extended with Maple code by R. H. Hardin and N. J. A. Sloane, Oct 18 2009

A152296 Let f(M,N)=2^(-M)*sum_{i=0..N} {(-1)^{i}M!N!(2M-2i)!}/{i!(M-i)!(N-i)!6^{N-i}}; then a(n) = f(3n,n).

Original entry on oeis.org

1, 6, 113400, 32901422400, 67651716132000000, 608762379843757339200000, 17903325789347617610786995200000, 1415199921956087613201896962521600000000, 261375521452474271183649591888039276441600000000, 101519644940256627137917269623207295713536128000000000000, 76392226231236455854222646891536244623780022885776896000000000000
Offset: 0

Views

Author

N. J. A. Sloane, Oct 18 2009

Keywords

Crossrefs

A variant of A132202. Cf. A134648, A134772, A152296.

Programs

  • Mathematica
    Table[2^(-3*n) * Sum[(-1)^i * (3*n)! * n! * (6*n-2*i)! / (i! * (3*n-i)! * (n-i)! * 6^(n-i)), {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Oct 21 2023 *)
Table[(6*n)! * Hypergeometric1F1[-n, 1/2 - 3*n, -3/2] / (2^(4*n) * 3^n), {n, 0, 20}] (* Vaclav Kotesovec, Oct 21 2023 *)

Formula

a(n) ~ sqrt(Pi) * 2^(2*n+1) * 3^(5*n + 1/2) * n^(6*n + 1/2) / exp(6*n + 1/2). - Vaclav Kotesovec, Oct 21 2023
Showing 1-3 of 3 results.

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

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