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A123244 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). +0
1
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; listen)
OFFSET

1,4

COMMENT

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

REFERENCES

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

LINKS

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

FORMULA

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

EXAMPLE

{1},

{1, 1}.

{3, 3, 1},

{15, 15, 9, 4},

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

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

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: A039797 A112292 A001497 this_sequence A105599 A106210 A033842

Adjacent sequences: A123241 A123242 A123243 this_sequence A123245 A123246 A123247

KEYWORD

nonn,tabl

AUTHOR

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

EXTENSIONS

Edited by njas, Oct 08 2006

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Last modified July 25 07:41 EDT 2008. Contains 142293 sequences.


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