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A065919 Bessel polynomial y_n(4). 6
1, 5, 61, 1225, 34361, 1238221, 54516085, 2836074641, 170218994545, 11577727703701, 880077524475821, 73938089783672665, 6803184337622361001, 680392371852019772765, 73489179344355757819621, 8525425196317119926848801, 1057226213522667226687070945 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Main diagonal of A143411. - Peter Bala, Aug 14 2008

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

LINKS

Harry J. Smith, Table of n, a(n) for n=0..100

W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

Index entries for sequences related to Bessel functions or polynomials

FORMULA

y_n(x) = sum ((n+k)!*(x/2)^k/((n-k)!*k!), k=0..n);

Recurrence relation: a(0) = 1, a(1) = 5, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A143412(n) satisfies the same recurrence relation. 1/sqrt(e) = 1 - 2*sum {n = 0..inf} (-1)^n/(a(n)*a(n+1)) = 1 - 2*(1/(1*5) - 1/(5*61) + 1/(61*1225) - ...). - Peter Bala, Aug 14 2008

G.f.: 1/Q(0), where Q(k)= 1 - x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013

a(n) = exp(1/4)/sqrt(2*Pi)*BesselK(n+1/2,1/4). - Gerry Martens, Jul 22 2015

a(n) ~ 2^(3*n+1/2) * n^n / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015

MATHEMATICA

Table[Sum[(n+k)!*2^k/((n-k)!*k!), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *)

PROG

(PARI) { for (n=0, 100, if (n>1, a=4*(2*n - 1)*a1 + a2; a2=a1; a1=a, if (n, a=a1=5, a=a2=1)); write("b065919.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 04 2009

(PARI) a(n) = sum(k=0, n, (n+k)!*2^k/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013

CROSSREFS

Cf. A001515, A001517, A001518.

Polynomial coefficients are in A001498.

A143411 (main diagonal), A143412. - Peter Bala, Aug 14 2008

Sequence in context: A217820 A217821 A009825 * A196125 A096537 A115047

Adjacent sequences:  A065916 A065917 A065918 * A065920 A065921 A065922

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 08 2001

EXTENSIONS

Recurrence relation a(2) = 5 corrected to a(1) = 5 by Harry J. Smith, Nov 04 2009

STATUS

approved

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Last modified December 4 19:31 EST 2016. Contains 278755 sequences.