A111601 Exponential (binomial) convolution of A001818 (with interspersed zeros) and A000142 (factorials).

1, 2, 9, 36, 225, 1350, 11025, 88200, 893025, 8930250, 108056025, 1296672300, 18261468225, 255660555150, 4108830350625, 65741285610000, 1187451971330625, 21374135483951250, 428670161650355625, 8573403233007112500

Offset: 1

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• nonn
• easy

Crossrefs

Second column (m=1) of triangle |A111595|.

Formula

E.g.f.: (1/sqrt(1-x^2))*x/(1-x).

a(n) = n!*Sum_{k=0..floor((n-1)/2)} binomial(2*k, k)/(4^k).

a(2*k+1) = A111595(2*k+2, 0)= ((2*k+1)!!)^2 = A001818(k+1), k >= 0.

D-finite with recurrence (-n+1)*a(n) +n*a(n-1) +n*(n-1)^2*a(n-2)=0. a(1)=1, a(2)=2. - Sergei N. Gladkovskii, Jul 27 2012, corrected Aug 11 2025