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 1-2 of 2 results.

A319743 Row sums of A174158.

Original entry on oeis.org

1, 2, 11, 74, 602, 5452, 53559, 558602, 6106034, 69298580, 811086718, 9740402476, 119550632788, 1495039156600, 19002275811887, 244983878813514, 3198363309664658, 42225545561470084, 563083734161627910, 7576864105285884420, 102790882283750139060, 1404908982711268821720
Offset: 1

Views

Author

Stefano Spezia, Dec 23 2018

Keywords

Links

Crossrefs

Cf. A000290, A007318, A174158, A174696.

Programs

  • GAP
    List([1..20], n->Sum([1..n], m->(Binomial(n-1, m-1)*Binomial(n, m-1)/m)^2));
  • Magma
    [(&+[(Binomial(n-1,j-1)*Binomial(n,j-1)/j)^2 : j in [1..n]]): n in [1..25]]; // G. C. Greubel, Feb 15 2021
  • Maple
    a := n -> add(binomial(n-1, m-1)^2*binomial(n, m-1)^2/m^2, m = 1 .. n): seq(a(n), n = 1 .. 20)
  • Mathematica
    Table[HypergeometricPFQ[{1-n,1-n,-n,-n},{1,2,2},1],{n,1,20}]
  • PARI
    a(n) = sum(m=1, n, (binomial(n-1, m-1)*binomial(n, m-1)/m)^2);
  • Sage
    [hypergeometric([-n, -n, -n+1, -n+1], [1, 2, 2], 1).simplify_hypergeometric() for n in (1..25)] # G. C. Greubel, Feb 15 2021

Formula

a(n) = Sum_{m=1..n} (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.
a(n) = Sum_{m=1..n} A000290(A007318(n - 1, m - 1)*A007318(n, m - 1)/m).
a(n) = 4F3([1 - n, 1 - n, - n, - n], [1, 2, 2], 1), where F is the generalized hypergeometric function.
From Vaclav Kotesovec, Dec 24 2018: (Start)
Recurrence: n*(n+1)^3*(5*n^2 - 10*n + 4)*a(n) = 2*n*(2*n - 1)*(15*n^4 - 30*n^3 + 7*n^2 + 8*n - 8)*a(n-1) + 4*(n-2)^2*(4*n - 5)*(4*n - 3)*(5*n^2 - 1)*a(n-2).
a(n) ~ 2^(4*n + 1/2) / (Pi^(3/2) * n^(7/2)).
(End)

A174694 Triangle T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 121, 121, 1, 1, 1081, 2281, 1081, 1, 1, 10081, 35281, 35281, 10081, 1, 1, 100801, 524161, 876961, 524161, 100801, 1, 1, 1088641, 7862401, 19716481, 19716481, 7862401, 1088641, 1, 1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1
Offset: 1

Views

Author

Roger L. Bagula, Mar 27 2010

Keywords

Examples

			Triangle begin as:
  1;
  1,        1;
  1,       13,         1;
  1,      121,       121,         1;
  1,     1081,      2281,      1081,         1;
  1,    10081,     35281,     35281,     10081,         1;
  1,   100801,    524161,    876961,    524161,    100801,         1;
  1,  1088641,   7862401,  19716481,  19716481,   7862401,   1088641,        1;
  1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1;

Links

Crossrefs

Cf. A000108, A174696, A176013.

Programs

  • Magma
    A174694:= func< n, k | (Factorial(n)/k)*Binomial(n-1, k-1)*Binomial(n, k-1) - Factorial(n) + 1 >;
[A174694(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021
  • Mathematica
    T[n_, k_]:= n!*(1/k)*Binomial[n-1, k-1]*Binomial[n, k-1] - n! + 1;
Table[T[n, k], {n,12}, {k,n}]//Flatten
  • Sage
    def A174694(n, k): return (factorial(n)/k)*binomial(n-1, k-1)*binomial(n, k-1) - factorial(n) + 1
flatten([[A174694(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021

Formula

T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1.
From G. C. Greubel, Feb 09 2021: (Start)
T(n, k) = (-1)^n * k! * A176013(n, k) - n! + 1.
Sum_{k=1..n} T(n,k) = n! * (C_{n} - n) + n, where C_{n} are the Catalan numbers (A000108). (End)

Extensions

Edited by G. C. Greubel, Feb 09 2021
Showing 1-2 of 2 results.

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