A194590 a(n) = (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2.

0, 0, 1, -2, 7, -14, 38, -76, 187, -374, 874, -1748, 3958, -7916, 17548, -35096, 76627, -153254, 330818, -661636, 1415650, -2831300, 6015316, -12030632, 25413342, -50826684, 106853668, -213707336, 447472972, -894945944, 1867450648, -3734901296, 7770342787

Offset: 0

Author

Peter Luschny, Aug 30 2011

Keywords

• sign

Comments

The binomial transform of a(n) are the complementary Riordan numbers A194589 (see link).

Links

• Peter Luschny, The lost Catalan numbers

Crossrefs

Cf. A107373 (has offset 1).

Programs

Maple

A056040 := n -> n!/iquo(n,2)!^2:
A152271 := n -> `if`(n mod 2 = 0, 1, (n+1)/2):
A194590 := n -> (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2:

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2;
a[n_] := (-1)^n (sf[n + 1] * If[EvenQ[n], 1, (n + 1)/2] - 2^n)/2;
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 26 2019 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*cr(k), where cr(k) are the complementary Riordan numbers A194589.