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A123244
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Triangle read by rows: coefficients of expansion in powers of x of the polynomials defined by p(k, x) = (2*k - 1)*p(k - 1, x) + (k - 1)^2*x^2*p(k - 2, x).
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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
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REFERENCES
| Abramowitz and Stegun, Handbook of Mathematical Functions,9th printing,1972, page 18 and page 22
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LINKS
| 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].
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FORMULA
| p(k, x) = (2*k - 1)*p(k - 1, x) + (k - 1)^2*x^2*p(k - 2, x)
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EXAMPLE
| {1},
{1, 1}.
{3, 3, 1},
{15, 15, 9, 4},
{105, 105, 90, 55, 9},
{945, 945, 1050, 735, 225, 64}
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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]
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CROSSREFS
| Cf. A000124, A001147, A000217, A006882.
Sequence in context: A143171 A112292 A001497 * A105599 A106210 A033842
Adjacent sequences: A123241 A123242 A123243 * A123245 A123246 A123247
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KEYWORD
| nonn,tabl
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AUTHOR
| Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 07 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 08 2006
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