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A048594
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Triangle a(n,k) = k! * Stirling1(n,k), 1<=k<=n.
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14
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1, -1, 2, 2, -6, 6, -6, 22, -36, 24, 24, -100, 210, -240, 120, -120, 548, -1350, 2040, -1800, 720, 720, -3528, 9744, -17640, 21000, -15120, 5040, -5040, 26136, -78792, 162456, -235200, 231840, -141120, 40320, 40320, -219168, 708744, -1614816, 2693880, -3265920, 2751840
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row sums (unsigned) give A007840(n), n>=1; (signed): A006252(n), n>=1.
Apart from signs, coefficients in expansion of n-th derivative of 1/log(x).
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FORMULA
| a(n, k)= k*a(n-1, k-1)-(n-1)*a(n-1, k) if n>=k>=1, a(n, 0) := 0 and a(1, 1)=1, else 0.
E.g.f. k-th column: log(1+x)^k, k>=1.
From Peter Bala, Nov 25 2011: (Start):
E.g.f.: 1/(1-t*log(1+x)) = 1 + t*x + (-t+2*t^2)*x^2/2! + ....
The row polynomials are given by D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator exp(-x)*d/dx.
(End)
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EXAMPLE
| Triangle begins
1;
-1, 2;
2, -6, 6;
-6, 22, -36, 24;
24, -100, 210, -240, 120; ...
The 2nd derivative of 1/log(x) is -2/x^3*log(x)^2-6/x^3*log(x)^3-6/x^3*log(x)^4.
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MAPLE
| with(combinat): A048594 := (n, k)->k!*stirling1(n, k);
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MATHEMATICA
| Flatten[Table[k!*StirlingS1[n, k], {n, 10}, {k, n}]] (* Harvey P. Dale, Aug 28 2011 *)
Join @@ CoefficientRules[ -Table[ D[ 1/Log[z], {z, n}], {n, 9}] /. Log[z] -> -Log[z], {1/z, 1/Log[z]}, "NegativeLexicographic"][[All, All, 2]] (* Oleg Marichev, oleg(AT)wolfram.com & Maxim Rytin, and m.r(AT)inbox.ru; submitted by RGWv, Aug 29 2011 *)
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PROG
| (PARI) {T(n, k)= if(k<1| k>n, 0, stirling(n, k)* k!)} /* Michael Somos Apr 11 2007 */
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CROSSREFS
| Cf. A008275, A019538, A075181.
Sequence in context: A084700 A122766 A033742 * A178801 A130493 A163118
Adjacent sequences: A048591 A048592 A048593 * A048595 A048596 A048597
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KEYWORD
| sign,tabl,easy,nice
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AUTHOR
| Oleg Marichev (oleg(AT)wolfram.com)
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