E.g.f.: 1 + x + 6*x^2/2!^2 + 63*x^3/3!^2 + 1162*x^4/4!^2 + 31263*x^5/5!^2 +...
such that A(x) = BesselI(0,2)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
Q(x) = 1/(1-x) + 1/((1-x)*(4-x^2)) + 1/((1-x)*(4-x^2)*(9-x^3)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)*(25-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients q(n) in Q(x) = Sum_{n>=0} q(n)*x^n/n!^2 begin:
q(0) = 1.279585302336067267437204440811533...
q(1) = 1.279585302336067267437204440811533...
q(2) = 5.397926511680336337186022204057666...
q(3) = 48.69967981446729610442301759976513...
q(4) = 789.3250187996735809262470013346725...
q(5) = 19745.00072507184117617488656759887...
q(6) = 713288.6822890207712374724807435860...
q(7) = 34956701.28771539805703277298850790...
q(8) = 2239176303.370447012433955813571405...
q(9) = 181385849371.3820539848573249577420...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A249078 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = BesselI(0,2)*1 - q(0) = 1;
a(1) = BesselI(0,2)*1 - q(1) = 1;
a(2) = BesselI(0,2)*5 - q(2) = 6;
a(3) = BesselI(0,2)*49 - q(3) = 63;
a(4) = BesselI(0,2)*856 - q(4) = 1162;
a(5) = BesselI(0,2)*22376 - q(5) = 31263;
a(6) = BesselI(0,2)*842536 - q(6) = 1207344;
a(7) = BesselI(0,2)*42409480 - q(7) = 61719326;
a(8) = BesselI(0,2)*2782192064 - q(8) = 4103067834; ...
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