A002695 P_n'(3), where P_n is n-th Legendre polynomial. (Formerly M4642 N1985)

1, 9, 66, 450, 2955, 18963, 119812, 748548, 4637205, 28537245, 174683718, 1064611782, 6464582943, 39132819495, 236256182280, 1423046656008, 8554078990377, 51327262010673, 307488810131530, 1839455028693450

Offset

1,2

References

H. Bateman, Some problems in potential theory, Messenger Math., 52 (1922), 71-78.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=1..100

H. Bateman, Some problems in potential theory, Messenger Math., 52 (1922), 71-78. [Annotated scanned copy]

John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.

Formula

G.f.: x*(1-6*x+x^2)^(-3/2). [corrected by Vaclav Kotesovec, Oct 04 2012]

a(n) = Gegenbauer_C(n,3/2,3). - Paul Barry, Apr 20 2009

D-finite with recurrence: -n*a(n-2) + 3*(2*n-1)*a(n-1) + (1-n)*a(n) = 0. - Vaclav Kotesovec, Oct 04 2012

a(n) ~ (3+2*sqrt(2))^n*sqrt(n)/(4*sqrt(2*Pi)*sqrt(3*sqrt(2)-4)). - Vaclav Kotesovec, Oct 04 2012

a(n) = (n+1) * n * A001003(n)/2, n>0. - Vladimir Kruchinin, Mar 29 2013

a(n) = Sum_{i=1..n+1} i*binomial(n+i+1,i)*binomial(n+1,i)/2. - Gerry Martens, Apr 08 2018

Mathematica

Table[SeriesCoefficient[x*(1-6x+x^2)^(-3/2), {x, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 04 2012 *)
a[n_]:= Sum[(i Binomial[n+i+1, i] Binomial[n+1, i]), {i, 1, n+1}]/2
Table[a[n], {n, 0, 20}] (* Gerry Martens, Apr 08 2018 *)

Prog

(PARI)
N = 66;  x = 'x + O('x^N);
gf = x*(1-6*x+x^2)^(-3/2);
Vec(gf)
/* Joerg Arndt, Mar 29 2013 */

Crossrefs

Cf. A001850.

Sequence in context: A279129 A051375 A081902 * A003408 A037698 A037607

Adjacent sequences: A002692 A002693 A002694 * A002696 A002697 A002698

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved