A337446 E.g.f.: exp(2*x) * (BesselI(0,2*x) - BesselI(1,2*x)) / (sec(x) + tan(x)).

1, 0, 1, 0, 3, -9, 5, -235, 35, -5939, 10773, -199746, 961521, -10506833, 82135911, -836458064, 8282576627, -90730736923, 1034615625645, -12538466040640, 159529541334325, -2133316798885373, 29875632576041747, -437461119834677379, 6683837093985315589

Offset: 0

Author

Ilya Gutkovskiy, Aug 27 2020

Keywords

• sign

Comments

Inverse boustrophedon transform of Catalan numbers.

Links

• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000108, A000111, A000753.

Programs

Mathematica

nmax = 24; CoefficientList[Series[Exp[2 x] (BesselI[0, 2 x] - BesselI[1, 2 x])/(Sec[x] + Tan[x]), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := CatalanNumber[n]; t[n_, k_] := t[n, k] = t[n, k - 1] - t[n - 1, n - k]; a[n_] := t[n, n]; Table[a[n], {n, 0, 24}]

Python

from itertools import islice, count, accumulate
from operator import sub
def A337446_gen(): # generator of terms
    blist, c = tuple(), 1
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=c)))[-1]
        c = c*(4*i+2)//(i+2)
A337446_list = list(islice(A337446_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000108(k) * A000111(n-k).