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A132393
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Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows.
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42
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1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0, 362880, 1026576, 1172700
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OFFSET
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0,8
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COMMENTS
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Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938 .
A094645*A007318 as infinite lower triangular matrices.
Row sums are the factorial numbers. - Roger L. Bagula, Apr 18 2008
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REFERENCES
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Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150
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LINKS
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Table of n, a(n) for n=0..58.
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FORMULA
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T(n,k)=T(n-1,k-1)+(n-1)*T(n-1,k), n,k>=1 ; T(n,0)=T(0,k) ; T(0,0)=1 .
Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively . - Philippe DELEHAM, Nov 13 2007
Expand 1/(1-t)^x = Sum[p(x,n)*t^n/n!,{n,0,Infinity}]; then the coefficients of the p(x,n) produce the triangle. - Roger L. Bagula, Apr 18 2008
Sum_{k=0..n}T(n,k)*2^k*x^(n-k) = A000142(n+1), A000165(n), A008544(n), A001813(n), A047055(n), A047657(n), A084947(n), A084948(n), A084949(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . [From Philippe DELEHAM, Sep 18 2008]
a(n)=Sum_{k=0..n}T(n,k)*3^k*x^(n-k)= A001710(n+2), A001147(n+1), A032031(n), A008545(n), A047056(n), A011781(n), A144739(n), A144756(n), A144758(n) for x=1,2,3,4,5,6,7,8,9,respectively . [From Philippe DELEHAM, Sep 20 2008]
Sum_{k=0..n}T(n,k)*4^k*x^(n-k)= A001715(n+3), A002866(n+1), A007559(n+1), A047053(n), A008546(n), A049308(n), A144827(n), A144828(n), A144829(n) for x=1,2,3,4,5,6,7,8,9 respectively . [From Philippe DELEHAM, Sep 21 2008]
Sum_{k, 0<=k<=n}x^k*T(n,k) = x*(1+x)*(2+x)*...*(n-1+x), n>=1. [From Philippe DELEHAM, Oct 17 2008]
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 1, 1;
0, 2, 3, 1;
0, 6, 11, 6, 1;
0, 24, 50, 35, 10, 1;
0, 120, 274, 225, 85, 15, 1 ;...
Production matrix is
0, 1
0, 1, 1
0, 1, 2, 1
0, 1, 3, 3, 1
0, 1, 4, 6, 4, 1
0, 1, 5, 10, 10, 5, 1
0, 1, 6, 15, 20, 15, 6, 1
0, 1, 7, 21, 35, 35, 21, 7, 1
...
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MAPLE
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a132393_row := proc(n) local k; seq(coeff(expand(pochhammer (x, n)), x, k), k=0..n) end: [Peter Luschny, Nov 28 2010]
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MATHEMATICA
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p[t_] = 1/(1 - t)^x; Table[ ExpandAll[(n!)SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[(n!)* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] - Roger L. Bagula, Apr 18 2008
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PROG
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(Maxima) create_list(abs(stirling1(n, k)), n, 0, 12, k, 0, n); /* Emanuele Munarini (Mar 11, 2011) */
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CROSSREFS
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Essentially a duplicate of A048994. Cf. A008275, A008277, A130534.
Sequence in context: A144633 A005210 A048994 * A121434 A137329 A171996
Adjacent sequences: A132390 A132391 A132392 * A132394 A132395 A132396
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe DELEHAM, Nov 10 2007, Oct 15 2008, Oct 17 2008
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STATUS
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approved
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