A259335 a(n) = ( Sum_{k=0..n} binomial(2*n, k)^2 * (binomial(2*n, k+1) - binomial(2*n, k-1)) )/(n*binomial(2*n, n)).

1, 7, 61, 611, 6686, 77729, 944245, 11859355, 152893720, 2013070126, 26967817306, 366542344117, 5043651762826, 70138959074461, 984384594022117, 13927418363218955, 198459156018467084, 2845950809029225472, 41044332341739034032, 594983281327999736694

Offset: 1

Author

N. J. A. Sloane, Jun 25 2015

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..835
• H. W. Gould, Problem E2384, Amer. Math. Monthly, 79 (1972), p. 1034.
• D. Zeitlin & N. J. A. Sloane, Correspondence, 1974 & 1991

Programs

Maple

f:=proc(n) local b;
b:=binomial;
add(b(2*n,k)^2*(b(2*n,k+1)-b(2*n,k-1)),k=0..n)/(n*b(2*n,n));
end;

PARI

a(n) = sum(k=0, n, k/(n+k)*binomial(n+k, k)^2)/(n+1); \\ Seiichi Manyama, Jul 16 2024

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (k/(n+k)) * binomial(n+k,k)^2. - Seiichi Manyama, Jul 16 2024