cp's OEIS Frontend

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).

%I A249607 #9 Nov 02 2014 04:16:58
%S A249607 1,1,4,37,600,15229,554868,27444786,1770376080,144306428161,
%T A249607 14507072762052,1762845211827574,254794661274061848,
%U A249607 43191427238728121445,8488249087135630544628,1914196040519793284483542,491024013925643339847990144,142153433027873627036756565313
%N 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).
%C A249607 Here BesselJ(0,2) = Sum_{n>=0} (-1)^n/n!^2 = 0.223890779141235668... (A091681).
%H A249607 Paul D. Hanna, <a href="/A249607/b249607.txt">Table of n, a(n) for n = 0..100</a>
%e A249607 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 +...
%e A249607 such that A(x) = BesselJ(0,2)*P(x) + Q(x), where
%e A249607 P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
%e A249607 Q(x) = Sum_{n>=1} -(-1)^n/Product_{k=1..n} (k^2 - x^k).
%e A249607 More explicitly,
%e A249607 P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
%e A249607 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)) -+...
%e A249607 We can illustrate the initial terms a(n) in the following manner.
%e A249607 The coefficients q(n) in Q(x) = Sum_{n>=0} q(n)*x^n/n!^2 begin:
%e A249607 q(0) = 0.776109220858764331948172545350051...
%e A249607 q(1) = 0.776109220858764331948172545350051...
%e A249607 q(2) = 2.880546104293821659740862726750256...
%e A249607 q(3) = 26.02935182207945226546045472215251...
%e A249607 q(4) = 408.3494930551022681476356988196439...
%e A249607 q(5) = 10219.21992593571069167230887475274...
%e A249607 q(6) = 366231.9585054598651822855036690508...
%e A249607 q(7) = 17949694.47982534876046938459857209...
%e A249607 q(8) = 1147468931.070477389192467314975593...
%e A249607 q(9) = 92955330843.11376518199210023477232...
%e A249607 and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
%e A249607 A249588 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
%e A249607 from which we can generate this sequence like so:
%e A249607 a(0) = BesselJ(0,2)*1 + q(0) = 1;
%e A249607 a(1) = BesselJ(0,2)*1 + q(1) = 1;
%e A249607 a(2) = BesselJ(0,2)*5 + q(2) = 4;
%e A249607 a(3) = BesselJ(0,2)*49 + q(3) = 37;
%e A249607 a(4) = BesselJ(0,2)*856 + q(4) = 600;
%e A249607 a(5) = BesselJ(0,2)*22376 + q(5) = 15229;
%e A249607 a(6) = BesselJ(0,2)*842536 + q(6) = 554868;
%e A249607 a(7) = BesselJ(0,2)*42409480 + q(7) = 27444786;
%e A249607 a(8) = BesselJ(0,2)*2782192064 + q(8) = 1770376080; ...
%o A249607 (PARI) \p100 \\ set precision
%o A249607 {P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
%o A249607 {Q=Vec(serlaplace(serlaplace(sum(n=1, 201, -(-1)^n*prod(k=1, n, 1./(k^2-x^k +O(x^31))))))); }
%o A249607 for(n=0, 30, print1(round(besselj(0,2)*P[n+1]+Q[n+1]), ", "))
%Y A249607 Cf. A249590, A249474, A249588, A091681.
%K A249607 nonn
%O A249607 0,3
%A A249607 _Paul D. Hanna_, Nov 02 2014