A000658 Strehl's sequence "C_n^(3)".

1, 4, 68, 1732, 51076, 1657904, 57793316, 2117525792, 80483121028, 3147565679824, 125937573689968, 5133632426499152, 212530848994367524, 8914634034287235856, 378138515326996979168, 16196097181014298854032

Offset: 0

Author

Don Knuth

Keywords

• nonn
• easy
• nice

References

• Volker Strehl, Binomial identities - combinatorial and algorithmic aspects. Trends in discrete mathematics. Discrete Math. 136 (1994), no. 1-3, 309-346.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..100
• Vaclav Kotesovec, Recurrence (of order 6)

Crossrefs

Third row of array A094424.

Cf. A007318.

Programs

Haskell

a000658 n = sum $ map c3 [0..n] where
   c3 k = (a007318' n k)^2 * (a007318' (2*k) k)^2 * a007318' (2*k) (n-k)
-- Reinhard Zumkeller, May 20 2015

Maple

A000658:=n->add(binomial(n,k)^2*binomial(2*k,k)^2*binomial(2*k,n-k), k=0..n): seq(A000658(n), n=0..15); # Wesley Ivan Hurt, Sep 19 2014

Mathematica

Table[Sum[Binomial[n,k]^2 Binomial[2k,k]^2 Binomial[2k,n-k],{k,0,n}], {n,0,25}] (* Harvey P. Dale, Oct 19 2011 *)

PARI

a(n)=sum(k=1,n, binomial(n, k)^2 * binomial(2k, k)^2 * binomial(2k, n-k)) \\ Charles R Greathouse IV, Sep 19 2014

Formula

Sum binomial(n, k)^2 * binomial(2k, k)^2 * binomial(2k, n-k); k=0..n.

a(n) ~ 7^(2*n+5/2) / (20 * sqrt(15) * Pi^2 * n^2). - Vaclav Kotesovec, Mar 09 2014