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A231894 Boustrophedon transform of the Catalan numbers A000108. 1
1, 3, 10, 37, 149, 648, 3039, 15401, 84619, 505500, 3287256, 23250514, 178382427, 1478782490, 13187788246, 125958159631, 1283067859947, 13886218459612, 159124624924418, 1924735353849082, 24506483918914367, 327627501208785322 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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).
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
Sequence in context: A151058 A044048 A192240 * A086444 A064613 A270789
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 18 2013
STATUS
approved

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Last modified June 21 21:50 EDT 2024. Contains 373559 sequences. (Running on oeis4.)