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A152818
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Array read by antidiagonals: T(n,k) = (k+1)^n*(n+k)!/n!.
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13
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1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.
Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. [From Omar E. Pol (info(AT)polprimos.com), Jan 06 2009]
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FORMULA
| E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2!+(1+12*t+18*t^2+6*t^3)*x^3/3!+.... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011
From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650. Row sums of triangle are A072597. - Peter Bala, Oct 09 2011
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EXAMPLE
| a(8)=18. Then a(6)+a(7)+a(8)+a(9)=A072597(3)=37.
Contribution from Omar E. Pol (info(AT)polprimos.com), Jan 06 2009: (Start)
T(2,3)=960 because (3+1)^2*(2+3)!/2! = 16*120/2 = 960.
Array begins:
1, 1, 2, 6, 24, 120,
1, 4, 18, 96, 600,
1, 12, 108, 960,
1, 32, 540,
1, 80,
1,
(End)
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MATHEMATICA
| len = 45; m = 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 + #2[[1]]^2 + (#2[[2]] - 1)*#2[[2]] + #2[[1]]*(2*#2[[2]] - 3))/ 2, #1}& , Table[(k + 1)^n*(n + k)!/n!, {n, 0, m}, {k, 0, m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-François Alcover, May 27 2011 *)
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CROSSREFS
| Cf. A000012, A000142, A001563, A001787, A006043, A006044, A055533. [From Omar E. Pol (info(AT)polprimos.com), Jan 06 2009]
Cf. A072597 (row sums of triangle), A152650, A154372.
Sequence in context: A049429 A183158 A174005 * A109244 A171650 A143777
Adjacent sequences: A152815 A152816 A152817 * A152819 A152820 A152821
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Dec 13 2008
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EXTENSIONS
| Better definition, extended and edited by Omar E. Pol (info(AT)polprimos.com) and N. J. A. Sloane (njas(AT)research.att.com), Jan 05 2009
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