1,4

Row sums: Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[p[n, x], x]]}], {n, 0, 15}] {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 (alex(AT)kolmogorov.com), Oct 08 2006

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

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].

p(k, x) = (2*k - 1)*p(k - 1, x) + (k - 1)^2*x^2*p(k - 2, x)

{1},

{1, 1}.

{3, 3, 1},

{15, 15, 9, 4},

{105, 105, 90, 55, 9},

{945, 945, 1050, 735, 225, 64}

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]

Cf. A000124, A001147, A000217, A006882.

Sequence in context: A143171 A112292 A001497 this_sequence A105599 A106210 A033842

Adjacent sequences: A123241 A123242 A123243 this_sequence A123245 A123246 A123247

nonn,tabl

Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 07 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 08 2006