A157100 Transform of Catalan numbers whose Hankel transform satisfies the Somos-4 recurrence.

1, 2, 3, 6, 14, 37, 105, 312, 956, 2996, 9554, 30897, 101083, 333947, 1112497, 3732956, 12605030, 42800318, 146046820, 500555448, 1722402304, 5948047170, 20607691518, 71610355541, 249520257107, 871614139397, 3051737703527

Offset: 0

Author

Paul Barry, Feb 22 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A157101.

The ratio of this generating function by the generating function of A025262 is x*(1-x), which means this sequence is the partial sums of A025262. - Sean A. Irvine, R. J. Mathar, Jun 27 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A157101.

Partial sums of A025262.

Programs

Mathematica

a[n_]:= Sum[(-1)^Binomial[k, 2]*Binomial[n-k, Floor[k/2]]*CatalanNumber[n-k], {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jan 11 2022 *)

Sage

def A157100(n): return sum((-1)^binomial(k,2)*binomial(n-k, k//2)*catalan_number(n-k) for k in (0..n))
[A157100(n) for n in (0..40)] # G. C. Greubel, Jan 11 2022

Formula

G.f.: (1+x)*c(x*(1-x^2)), c(x) the g.f. of A000108;

a(n) = Sum_{k=0..n} (-1)^binomial(n-k,2)*binomial(k,floor((n-k)/2))*A000108(k).

Conjecture: (n+1)*a(n) +(-5*n+1)*a(n-1) +2*(2*n-1)*a(n-2) +2*(2*n-7)*a(n-3) +2*(-2*n+7)*a(n-4) = 0. - R. J. Mathar, Feb 05 2015