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

0,2

Links

Table of n, a(n) for n=0..21.

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).

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

Example

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

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

Prog

(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

Crossrefs

Cf. A000108, A000736, A000753, A109449.

Sequence in context: A151058 A044048 A192240 * A086444 A064613 A270789

Adjacent sequences: A231891 A231892 A231893 * A231895 A231896 A231897

Keyword

nonn

Author

N. J. A. Sloane, Nov 18 2013

Status

approved