0,2

Numbers appearing as coefficients in the series expansion of an elliptic integral of the second kind. Defining f(x; c) = [1 - c^2sin^2(x)]^(1/2), consider the function E(c) obtained by integrating f(x; c) with respect to x between 0 and pi/2. E(c) is transformed and written as a power series in c (through an intermediate variable) which acts as a generating function for the sequence.

E'(k) is complete elliptic integral of second kind evaluated at k'. - Michael Somos, Mar 04 2003

A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between two recent integer sequences, Congressus Numerantium, 169 (2004), 79-99.

A. F. Jarvis, P. J. Larcombe and D. R. French, Power series identities generated by two recent integer sequences, Bulletin ICA, 43 (2005), 85-95.

P. J. Larcombe, A new asymptotic relation between two recent integer sequences, Congressus Numerantium, 175 (2005), 111-116.

P. J. Larcombe, D. R. French and E. J. Fennessey, The Fennessey-Larcombe-French sequence {1, 8, 144, 2432, 40000, ...}: formulation and asymptotic form, Congressus Numerantium, 158 (2002), 179-190.

P. J. Larcombe, D. R. French and E. J. Fennessey, The Fennessey-Larcombe-French sequence {1, 8, 144, 2432, 40000, ...}: a recursive formulation and prime factor decomposition, Congressus Numerantium, 160 (2003), 129-137.

T. D. Noe, Table of n, a(n) for n=0..200

a(n) = 8^n * 4F3( [5/4, 1/2, (1/2)-n/2, -n/2], [1, 1, 1/4] | 1 ).

G.f.: F(-1/2, 1/2;1;32x-256x^2)/(1-16x) = E'(1-16x)/(pi/2(1-16x)). - Michael Somos, Mar 04 2003

a(n)(n^3-n^2)=a(n-1)(8-32n^2+24n^3)+a(n-2)(256n^2-128n^3). - Michael Somos, Mar 04 2003

a(n) = 2^n*Sum_{k=0..n} (4*k^2-2*k-1)/(2*k-1)*binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 02 2005

E.g.f.: exp(8*x)*BesselI(0, 4*x)*(BesselI(0, 4*x)+16*x*BesselI(1, 4*x)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 02 2005

(PARI) a(n)=local(A); if(n<0, 0, A=agm(1, 1-16*x+x*O(x^n)); polcoeff((1-16*x-2*x*(1-8*x)*log(A)')/A, n))

(PARI) a(n)=if(n<0, 0, polcoeff(sum(k=0, n, binomial(2*k, k)^2/(1-2*k)*(2*x-16*x^2)^k, x*O(x^n))/(1-16*x), n))

Cf. A053175, A010370.

Sequence in context: A061996 A090931 A112464 this_sequence A061899 A134492 A067421

Adjacent sequences: A065406 A065407 A065408 this_sequence A065410 A065411 A065412

nonn,nice

P.J.Larcombe(AT)derby.ac.uk, Nov 14 2001