A249607 E.g.f.: BesselJ(0,2)*P(x) + Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) and Q(x) = Sum_{n>=1} -(-1)^n/Product_{k=1..n} (k^2 - x^k).

1, 1, 4, 37, 600, 15229, 554868, 27444786, 1770376080, 144306428161, 14507072762052, 1762845211827574, 254794661274061848, 43191427238728121445, 8488249087135630544628, 1914196040519793284483542, 491024013925643339847990144, 142153433027873627036756565313

Offset: 0

Author

Paul D. Hanna, Nov 02 2014

Keywords

• nonn

Comments

Here BesselJ(0,2) = Sum_{n>=0} (-1)^n/n!^2 = 0.223890779141235668... (A091681).

Examples

			E.g.f.: 1 + x + 4*x^2/2!^2 + 37*x^3/3!^2 + 600*x^4/4!^2 + 15229*x^5/5!^2 +...
such that A(x) = BesselJ(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)^n/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) = 0.776109220858764331948172545350051...
q(1) = 0.776109220858764331948172545350051...
q(2) = 2.880546104293821659740862726750256...
q(3) = 26.02935182207945226546045472215251...
q(4) = 408.3494930551022681476356988196439...
q(5) = 10219.21992593571069167230887475274...
q(6) = 366231.9585054598651822855036690508...
q(7) = 17949694.47982534876046938459857209...
q(8) = 1147468931.070477389192467314975593...
q(9) = 92955330843.11376518199210023477232...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A249588 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = BesselJ(0,2)*1 + q(0) = 1;
a(1) = BesselJ(0,2)*1 + q(1) = 1;
a(2) = BesselJ(0,2)*5 + q(2) = 4;
a(3) = BesselJ(0,2)*49 + q(3) = 37;
a(4) = BesselJ(0,2)*856 + q(4) = 600;
a(5) = BesselJ(0,2)*22376 + q(5) = 15229;
a(6) = BesselJ(0,2)*842536 + q(6) = 554868;
a(7) = BesselJ(0,2)*42409480 + q(7) = 27444786;
a(8) = BesselJ(0,2)*2782192064 + q(8) = 1770376080; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A249590, A249474, A249588, A091681.

Programs

PARI

\p100 \\ set precision
{P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
{Q=Vec(serlaplace(serlaplace(sum(n=1, 201, -(-1)^n*prod(k=1, n, 1./(k^2-x^k +O(x^31))))))); }
for(n=0, 30, print1(round(besselj(0,2)*P[n+1]+Q[n+1]), ", "))