The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 3
 1, 1, 4, 37, 600, 15229, 554868, 27444786, 1770376080, 144306428161, 14507072762052, 1762845211827574, 254794661274061848, 43191427238728121445, 8488249087135630544628, 1914196040519793284483542, 491024013925643339847990144, 142153433027873627036756565313 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Here BesselJ(0,2) = Sum_{n>=0} (-1)^n/n!^2 = 0.223890779141235668... (A091681). LINKS Paul D. Hanna, Table of n, a(n) for n = 0..100 EXAMPLE 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; ... PROG (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]), ", ")) CROSSREFS Cf. A249590, A249474, A249588, A091681. Sequence in context: A345102 A036245 A133471 * A097998 A352470 A177385 Adjacent sequences: A249604 A249605 A249606 * A249608 A249609 A249610 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 02 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 22 00:10 EDT 2023. Contains 365503 sequences. (Running on oeis4.)