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A000736
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Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ...
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4
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1, 2, 4, 10, 32, 120, 513, 2455, 13040, 76440, 492231, 3465163, 26530503, 219754535, 1959181266, 18710532565, 190588702776, 2062664376064, 23636408157551, 285900639990875, 3640199365715769, 48665876423760247
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OFFSET
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0,2
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LINKS
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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).
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FORMULA
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E.g.f.: (sec(x) + tan(x))*(integral(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)),x)+1). - Sergei N. Gladkovskii, Oct 30 2014
a(n) ~ n! * (6/Pi+2*exp(Pi)*((2-1/Pi)*BesselI(0,Pi)-2*BesselI(1,Pi))) * 2^n / Pi^n. - Vaclav Kotesovec, Oct 30 2014
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MAPLE
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egf := (sec(x/2)+tan(x/2))*(exp(x)*((x-1/2)*BesselI(0, x)-x*BesselI(1, x))+3/2);
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MATHEMATICA
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CoefficientList[Series[1/2*(3 + E^(2*x)*((4*x-1)*BesselI[0, 2*x] - 4*x*BesselI[1, 2*x]))*(Sec[x] + Tan[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 30 2014, after Peter Luschny *)
t[n_, 0] := If[n == 0, 1, CatalanNumber[n - 1]]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)
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PROG
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(Haskell)
a000736 n = sum $ zipWith (*) (a109449_row n) (1 : a000108_list)
(Python)
from itertools import accumulate, count, islice
def A000736_gen(): # generator of terms
yield 1
blist, c = (1, ), 1
for i in count(0):
yield (blist := tuple(accumulate(reversed(blist), initial=c)))[-1]
c = c*(4*i+2)//(i+2)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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