%I #20 Feb 13 2022 23:35:56
%S 1,1,1,3,3,1,15,15,9,4,105,105,90,55,9,945,945,1050,735,225,64,10395,
%T 10395,14175,10710,4725,2079,225,135135,135135,218295,173250,99225,
%U 53487,11025,2304,2027025,2027025,3783780,3108105,2182950,1327095
%N Triangle read by rows: coefficients of expansion in powers of x of the polynomials defined by p(n, x) = (2*n - 1)*p(n - 1, x) + (n - 1)^2*x^2*p(n - 2, x).
%C Row sums are r(n) = 1, 2, 7, 43, 364, 3964, 52704, 827856, 15000336, 307988496, 7066808640, 179201831040, 4976725959360, 150223212653760, 4897093428783360, 171459459114854400, ...
%C The first two terms in each row are identical double factorials for n>1, or a(n(n+1)/2 + 1) = a(n(n+1)/2 + 2) = (2n-1)!! for n>0. a(A000124[n]) = a(A000124[n]+1) = A001147[n] for n>0. Also the last term in each row is a square of double factorial a(n(n+1)/2) = (n-2)!!^2 for n>1. a(A000217[n]) = A006882[n-2]^2 for n>1. - _Alexander Adamchuk_, Oct 08 2006, for the old offset.
%C Row sums appear to obey r(n) +(-2*n+1)*r(n-1) -(n-1)^2*r(n-2)=0. - _R. J. Mathar_, Oct 05 2014
%D Abramowitz and Stegun, Handbook of Mathematical Functions,9th printing,1972, page 18 and page 22
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F T(n,k) = [x^k] p(n,x). p(0,x)=1. p(1,x)=1+x.
%F T(n,k) = (2n-1)*T(n-1,k)+(n-1)^2*T(n-2,k-2). T(n,k)=0 if k>n or k<0. T(0,0)=T(1,0)=T(1,1)=1.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 3, 3, 1;
%e 15, 15, 9, 4;
%e 105, 105, 90, 55, 9;
%e 945, 945, 1050, 735, 225, 64;
%e 10395, 10395, 14175, 10710, 4725, 2079, 225;
%e 135135, 135135, 218295, 173250, 99225, 53487, 11025, 2304;
%e 2027025, 2027025, 3783780, 3108105, 2182950, 1327095, ...;
%p A123244p := proc(n)
%p if n = 0 then
%p 1 ;
%p elif n = 1 then
%p 1+x ;
%p else
%p (2*n-1)*procname(n-1)+(n-1)^2*x^2*procname(n-2) ;
%p expand(%) ;
%p end if;
%p end proc:
%p A123244 := proc(n,k)
%p coeff( A123244p(n),x,k) ;
%p end proc:
%p seq( seq(A123244(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Oct 05 2014
%t p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = (2*k - 1)*p[k - 1, x] + (k - 1)^2*x^2*p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
%Y Cf. A000124, A001147, A000217, A006882.
%K nonn,tabl,easy
%O 0,4
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 07 2006
%E Edited by _N. J. A. Sloane_, Oct 08 2006
%E Offset set to 0 by _R. J. Mathar_, Oct 05 2014