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.

A137942 First bisection of A134772.

Original entry on oeis.org

1, 28, 27165600, 1445549490000000, 1081114481157129619200000, 5873237165016878140678626432000000, 156064894765355001368149078831725782016000000, 15583529649395480761968847415068808311749204480000000000, 4843348111055914672023195506389150149608445774198528000000000000000, 4067688449094150594904537709530563016131839124729830583634193326080000000000000
Offset: 0

Views

Author

N. J. A. Sloane, Oct 18 2009

Keywords

Links

Crossrefs

Cf. A134772.

Programs

  • Magma
    B:=Binomial; F:=Factorial;
A137942:= func< n | F(8*n)/(24)^(2*n)*(&+[B(2*n, j)*B(4*n, j)*(-6)^j/(F(j)*B(2*j, j)*B(8*n, 2*j)) : j in [0..2*n]]) >;
[A137942(n): n in [0..30]]; // G. C. Greubel, Oct 16 2023
  • Mathematica
    Table[((8*n)!/(24)^(2*n))*Hypergeometric1F1[-2*n,1/2-4*n,-3/2], {n,0,30}] (* G. C. Greubel, Oct 16 2023 *)
  • SageMath
    b=binomial; f=factorial;
def A137942(n): return (f(8*n)/(24)^(2*n))*sum(b(2*n,j)*b(4*n,j)*(-6)^j/(f(j)*b(2*j,j)*b(8*n,2*j)) for j in range(2*n+1))
[A137942(n) for n in range(31)] # G. C. Greubel, Oct 16 2023

Formula

From G. C. Greubel, Oct 16 2023: (Start)
a(n) = ((8*n)!/(24)^(2*n))*Sum_{j=0..2*n} ( b(2*n, j)*b(4*n, j)*(-6)^j )/( j! * b(2*j, j)*b(8*n, 2*j) ), where b(x,y) = binomial(x,y).
a(n) = ((8*n)!/(24)^(2*n))*Hypergeometric1F1([-2*n], [1/2-4*n], -3/2). (End)
a(n) ~ sqrt(Pi) * 2^(18*n + 2) * n^(8*n + 1/2) / (3^(2*n) * exp(8*n + 3/4)). - Vaclav Kotesovec, Oct 21 2023

A144649 Second bisection of A134772.

Original entry on oeis.org

0, 14400, 134289792000, 29865588836219136000, 64007711015400701105356800000, 799901135455942846519287494400000000000, 42346525471797343063631567858734790430720000000000, 7611746717262781749937067971966455935937523732684800000000000, 3949387898792061570875758855816554982971495343701121923966566400000000000
Offset: 0

Views

Author

N. J. A. Sloane, Oct 18 2009

Keywords

Links

Crossrefs

Cf. A134772.

Programs

  • Magma
    B:=Binomial; F:=Factorial;
A134772:= func< n | F(4*n)/(24)^n *(&+[B(n, j)*B(2*n, j)*(-6)^j/(F(j)*B(2*j, j)*B(4*n, 2*j)) : j in [0..n]]) >;
A144649:= func< n | A134772(2*n+1) >;
[A144649(n): n in [0..20]]; // G. C. Greubel, Oct 13 2023
  • Mathematica
    A134772[n_]:= ((4*n)!/(24)^n)*Hypergeometric1F1[-n,1/2-2*n,-3/2];
A144549[n_]:= A134772[2*n+1];
Table[A144549[n], {n,0,20}] (* G. C. Greubel, Oct 13 2023 *)
  • SageMath
    def A134772(n): return (factorial(4*n)/(24)^n)* simplify(hypergeometric([-n], [1/2-2*n], -3/2))
def A144649(n): return A134772(2*n+1)
[A144649(n) for n in range(21)] # G. C. Greubel, Oct 13 2023

Formula

a(n) = A134772(2*n+1). - G. C. Greubel, Oct 13 2023
a(n) ~ sqrt(Pi) * 2^(18*n + 11) * n^(8*n + 9/2) / (3^(2*n+1) * exp(8*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 1-4 of 4 results.

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

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