A002695 - OEIS

%I M4642 N1985 #46 Sep 28 2024 07:36:30

%S 1,9,66,450,2955,18963,119812,748548,4637205,28537245,174683718,

%T 1064611782,6464582943,39132819495,236256182280,1423046656008,

%U 8554078990377,51327262010673,307488810131530,1839455028693450

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

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

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

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

%H T. D. Noe, <a href="/A002695/b002695.txt">Table of n, a(n) for n=1..100</a>

%H H. Bateman, <a href="/A002692/a002692.pdf">Some problems in potential theory</a>, Messenger Math., 52 (1922), 71-78. [Annotated scanned copy]

%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.

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

%F a(n) = Gegenbauer_C(n,3/2,3). - _Paul Barry_, Apr 20 2009

%F 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

%F a(n) ~ (3+2*sqrt(2))^n*sqrt(n)/(4*sqrt(2*Pi)*sqrt(3*sqrt(2)-4)). - _Vaclav Kotesovec_, Oct 04 2012

%F a(n) = (n+1) * n * A001003(n)/2, n>0. - _Vladimir Kruchinin_, Mar 29 2013

%F a(n) = Sum_{i=1..n} i*binomial(n+i,i)*binomial(n,i)/2. - _Gerry Martens_, Apr 08 2018

%t Table[SeriesCoefficient[x*(1-6x+x^2)^(-3/2),{x,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 04 2012 *)

%t a[n_]:= Sum[(i Binomial[n+i+1,i] Binomial[n+1,i]),{i,1,n+1}]/2

%t Table[a[n], {n, 0, 20}] (* _Gerry Martens_, Apr 08 2018 *)

%o (PARI)

%o N = 66;  x = 'x + O('x^N);

%o gf = x*(1-6*x+x^2)^(-3/2);

%o Vec(gf)

%o /* _Joerg Arndt_, Mar 29 2013 */

%Y Cf. A001850.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, _Simon Plouffe_