A191349 Binomial sums: a(n) = Sum_{k=0..floor(n/3)} binomial(n-k,2*k)^2.

1, 1, 1, 2, 10, 37, 102, 251, 667, 2011, 6246, 18686, 54183, 157438, 466579, 1400509, 4202829, 12556360, 37480728, 112227269, 337221408, 1014992763, 3055963443, 9203918219, 27745375212, 83736861040, 252968402935, 764733200716, 2313088202001, 7000425465331

Offset

0,4

Links

Table of n, a(n) for n=0..29.

Formula

a(n) = sum(binomial(2*k,k)*sum(binomial(n+k-i,2*k)*binomial(n-k-i,2*i)*(-1)^(n-k-i),i=0..floor((n-k)/3)),k=0..n).

G.f.: sqrt((1-2*x+x^2+x^3+sqrt(1-4*x+6*x^2-6*x^3-11*x^4-2*x^5+x^6))/(2*(1-4*x+6*x^2-6*x^3-11*x^4-2*x^5+x^6))).

Mathematica

Table[Sum[Binomial[n-k, 2k]^2, {k, 0, Floor[n/3]}], {n, 0, 100}]
a[n_] := HypergeometricPFQ[ ({1, 1, 2, 2, 0, 0}-n)/3, {1/2, 1/2, 1, -n, -n}, 729/16]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Feb 22 2013 *)

Prog

(Maxima) makelist(sum(binomial(n-k, 2*k)^2, k, 0, floor(n/3)), n, 0, 26);

Crossrefs

Sequence in context: A144895 A236767 A154323 * A073110 A034547 A246604

Adjacent sequences: A191346 A191347 A191348 * A191350 A191351 A191352

Keyword

nonn,easy

Author

Emanuele Munarini, May 31 2011

Status

approved