

A238390


E.g.f.: x / BesselJ(1, 2*x) (even powers only).


7



1, 1, 4, 35, 546, 13482, 485892, 24108513, 1576676530, 131451399794, 13609184032808, 1712978776719938, 257612765775847132, 45620136452519144700, 9396239458048330569840, 2227147531572856811691105, 601916577165056911293330930, 183994483721828524163677628370
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

Aerated, the e.g.f. is e^(a.t) = 1/AC(i*t) = 1/[I_1(2i*t)/(it)] = 1/Sum_{n>=0} (1)^n t^(2n) / [n!(n+1)!] = a_0 + a_2 t^2/2! + a_4 t^4/4! + ... = 1 + t^2/2! + 4 t^4/4! + 35 t^6/6! + ..., where AC(t) is the e.g.f. for the aerated Catalan numbers c_n of A126120 and I_n(t) are the modified Bessel functions of the first kind (i = sqrt(1)). The signed, aerated sequence b_n = (i)^n a_n has the e.g.f. e^(b.t) = 1/AC(t) and, therefore, (i*a. + c.)^n = Sum_{k=0..n} binomial(n,k) i^k a_k c_(nk) vanishes except for n=0 for which it's unity.  Tom Copeland, Jan 23 2016
With q(n) = A126120(n+1) and q(0) = 0, d(2n) = (1)^n A238390(n) and zero for odd arguments, and r(2n+1) = (1)^n A180874(n+1) and zero for even arguments, then r(n) = (q. + d.)^n = Sum_{k=0..n} binomial(n,k) q(k) d(nk), relating these sequences (and A000108) through binomial convolutions. Then also, (r. + c. + d.)^n = r(n). See A180874 for proofs and for relations to A097610. For quick reference, q = (0, 1, 0, 2, 0, 5, 0, 14, ..), d = (1, 0, 1, 0, 4, 0, 35, 0, ..), and r = (0, 1, 0, 1, 0, 5, 0, 56, ..).  Tom Copeland, Jan 28 2016
Aerated and signed, this sequence contains the moments m(n) of the Appell polynomial sequence UMT(n,h1,h2) that is the umbral compositional inverse of the Appell sequence of Motzkin polynomials MT(n,h1,h2) of A097610 with exp[x UMT(.,h1,h2)] = e^(x*h1) / AC(x*y) where y = sqrt(h2) and AC is defined above. UMT(n,h1,h2) = (m.y + h1)^n with (m.)^(2n) = m(2n) = (1)^n A238390(n) and zero otherwise. Consequently, the associated lower triangular matrices A007318(n,k)*m(nk) and A007318(n,k)*A126120(nk) form an inverse pair (cf. also A133314), and MT(n,UMT(.,h1,h2),h2) = h1^n = UMT(n,MT(.,h1,h2),h2).  Tom Copeland, Jan 30 2016


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..98


FORMULA

a(n) ~ c * (n!)^2 / (sqrt(n) * r^n), where r = BesselJZero[1, 1]^2/16 = 0.91762316513274332857623611, and c = 1/(Sqrt[Pi]*BesselJ[2, BesselJZero[1, 1]]) = 1.4008104828035425937394082168...  Vaclav Kotesovec, Mar 01 2014, updated Apr 01 2018


MAPLE

S:= series(x/BesselJ(1, 2*x), x, 102):
seq((2*j)!*coeff(S, x, 2*j), j=0..50); # Robert Israel, Jan 31 2016


MATHEMATICA

Table[(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 40}], x] * Range[0, 40]!)[[n]], {n, 1, 41, 2}]


CROSSREFS

Cf. A103365, A180874, A115369, A000275, A002190.
Cf. A000108, A007318, A097610, A126120, A133314.
Sequence in context: A165933 A005973 A007134 * A251591 A125798 A129581
Adjacent sequences: A238387 A238388 A238389 * A238391 A238392 A238393


KEYWORD

nonn


AUTHOR

Vaclav Kotesovec, Mar 01 2014


STATUS

approved



