A123244 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).

1, 1, 1, 3, 3, 1, 15, 15, 9, 4, 105, 105, 90, 55, 9, 945, 945, 1050, 735, 225, 64, 10395, 10395, 14175, 10710, 4725, 2079, 225, 135135, 135135, 218295, 173250, 99225, 53487, 11025, 2304, 2027025, 2027025, 3783780, 3108105, 2182950, 1327095

Offset

0,4

Comments

Row sums are r(n) = 1, 2, 7, 43, 364, 3964, 52704, 827856, 15000336, 307988496, 7066808640, 179201831040, 4976725959360, 150223212653760, 4897093428783360, 171459459114854400, ...

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.

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

References

Abramowitz and Stegun, Handbook of Mathematical Functions,9th printing,1972, page 18 and page 22

Links

Table of n, a(n) for n=0..41.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Formula

T(n,k) = [x^k] p(n,x). p(0,x)=1. p(1,x)=1+x.

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.

Example

Triangle begins:
        1;
        1,       1;
        3,       3,       1;
       15,      15,       9,       4;
      105,     105,      90,      55,       9;
      945,     945,    1050,     735,     225,      64;
    10395,   10395,   14175,   10710,    4725,    2079,   225;
   135135,  135135,  218295,  173250,   99225,   53487, 11025, 2304;
  2027025, 2027025, 3783780, 3108105, 2182950, 1327095, ...;

Maple

A123244p := proc(n)
    if n = 0 then
        1 ;
    elif n = 1 then
        1+x ;
    else
        (2*n-1)*procname(n-1)+(n-1)^2*x^2*procname(n-2) ;
        expand(%) ;
    end if;
end proc:
A123244 := proc(n, k)
    coeff( A123244p(n), x, k) ;
end proc:
seq( seq(A123244(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Oct 05 2014

Mathematica

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]

Crossrefs

Cf. A000124, A001147, A000217, A006882.

Sequence in context: A143171 A112292 A001497 * A364505 A105599 A239895

Adjacent sequences: A123241 A123242 A123243 * A123245 A123246 A123247

Keyword

nonn,tabl,easy

Author

Roger L. Bagula and Gary W. Adamson, Oct 07 2006

Extensions

Edited by N. J. A. Sloane, Oct 08 2006

Offset set to 0 by R. J. Mathar, Oct 05 2014

Status

approved