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

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

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A001850.

Programs

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 *)

PARI

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

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} i*binomial(n+i,i)*binomial(n,i)/2. - Gerry Martens, Apr 08 2018