cp's OEIS Frontend

A249590 E.g.f.: BesselI(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/Product_{k=1..n} (k^2 - x^k).

%I A249590 #12 Nov 02 2014 04:23:58
%S A249590 1,1,6,63,1162,31263,1207344,61719326,4103067834,341454828363,
%T A249590 34946904263560,4304483416099530,629558493157805370,
%U A249590 107728435291299602135,21346960361800584031800,4847223770735591212039818,1250978551922243595690043914,364052135715732457875255719691
%N A249590 E.g.f.: BesselI(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/Product_{k=1..n} (k^2 - x^k).
%C A249590 Here BesselI(0,2) = Sum_{n>=0} 1/n!^2 = 2.2795853023360672... (A070910).
%H A249590 Paul D. Hanna, <a href="/A249590/b249590.txt">Table of n, a(n) for n = 0..100</a>
%e A249590 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 +...
%e A249590 such that A(x) = BesselI(0,2)*P(x) - Q(x), where
%e A249590 P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
%e A249590 Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).
%e A249590 More explicitly,
%e A249590 P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
%e A249590 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 A249590 We can illustrate the initial terms a(n) in the following manner.
%e A249590 The coefficients q(n) in Q(x) = Sum_{n>=0} q(n)*x^n/n!^2 begin:
%e A249590 q(0) = 1.279585302336067267437204440811533...
%e A249590 q(1) = 1.279585302336067267437204440811533...
%e A249590 q(2) = 5.397926511680336337186022204057666...
%e A249590 q(3) = 48.69967981446729610442301759976513...
%e A249590 q(4) = 789.3250187996735809262470013346725...
%e A249590 q(5) = 19745.00072507184117617488656759887...
%e A249590 q(6) = 713288.6822890207712374724807435860...
%e A249590 q(7) = 34956701.28771539805703277298850790...
%e A249590 q(8) = 2239176303.370447012433955813571405...
%e A249590 q(9) = 181385849371.3820539848573249577420...
%e A249590 and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
%e A249590 A249078 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
%e A249590 from which we can generate this sequence like so:
%e A249590 a(0) = BesselI(0,2)*1 - q(0) = 1;
%e A249590 a(1) = BesselI(0,2)*1 - q(1) = 1;
%e A249590 a(2) = BesselI(0,2)*5 - q(2) = 6;
%e A249590 a(3) = BesselI(0,2)*49 - q(3) = 63;
%e A249590 a(4) = BesselI(0,2)*856 - q(4) = 1162;
%e A249590 a(5) = BesselI(0,2)*22376 - q(5) = 31263;
%e A249590 a(6) = BesselI(0,2)*842536 - q(6) = 1207344;
%e A249590 a(7) = BesselI(0,2)*42409480 - q(7) = 61719326;
%e A249590 a(8) = BesselI(0,2)*2782192064 - q(8) = 4103067834; ...
%o A249590 (PARI) \p100 \\ set precision
%o A249590 {P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
%o A249590 {Q=Vec(serlaplace(serlaplace(sum(n=1, 201, prod(k=1, n, 1./(k^2-x^k +O(x^31))))))); }
%o A249590 for(n=0, 30, print1(round(besseli(0,2)*P[n+1]-Q[n+1]), ", "))
%Y A249590 Cf. A249607, A249592, A249588, A249078, A070910.
%K A249590 nonn
%O A249590 0,3
%A A249590 _Paul D. Hanna_, Nov 01 2014