A231894 Boustrophedon transform of the Catalan numbers A000108.

1, 3, 10, 37, 149, 648, 3039, 15401, 84619, 505500, 3287256, 23250514, 178382427, 1478782490, 13187788246, 125958159631, 1283067859947, 13886218459612, 159124624924418, 1924735353849082, 24506483918914367, 327627501208785322

Offset: 0

Author

N. J. A. Sloane, Nov 18 2013

Keywords

• nonn

Examples

			G.f. = 1 + 3*x + 10*x^2 +37*x^3 + 149*x^4 + 648*x^5 + 3039*x^6 + 15401*x^7 + ...
		

Links

• D. Berry, J. Broom, D. Dixon, A. Flaherty, Umbral Calculus and the Boustrophedon Transform, 2013
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

Crossrefs

Cf. A000108, A000736, A000753, A109449.

Programs

Maple

A000111 := proc(n)
    option remember;
    sec(x)+tan(x) ;
    coeftayl(%,x=0,n)*n! ;
end proc:
A109449 := proc(n,k)
    binomial(n,k)*A000111(n-k) ;
end proc:
A231894 := proc(n)
    add( A109449(n,k)*A000108(k+1),k=0..n) ;
end proc:
seq(A231894(n),n=0..30) ; # R. J. Mathar, Oct 04 2014

Python

from itertools import accumulate, count, islice
def A231894_gen(): # generator of terms
    blist, c = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c = c*(4*i+2)//(i+2)
A231894_list = list(islice(A231894_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A000108(k+1). - Philippe Deléham, Nov 20 2013

E.g.f.: exp(2*x)*I_1(2*x)*(sec(x)+tan(x))/x, where I_1(2*x) is the modified Bessel function of the first kind. - Sergei N. Gladkovskii, Nov 19 2014

a(n) ~ n! * exp(Pi) * BesselI(1, Pi) * 2^(n+3) / Pi^(n+2). - Vaclav Kotesovec, Jun 12 2015