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A136124
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Triangle read by rows: T(n,k)=(-1)^(n+k)*Sum(s(n,j),j=1..k), where s(n,j) are the signed Stirling numbers of the first kind (n>=2; 1<=k<=n-1; s(n,j)=A008275(n,j)).
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1
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1, 2, 1, 6, 5, 1, 24, 26, 9, 1, 120, 154, 71, 14, 1, 720, 1044, 580, 155, 20, 1, 5040, 8028, 5104, 1665, 295, 27, 1, 40320, 69264, 48860, 18424, 4025, 511, 35, 1, 362880, 663696, 509004, 214676, 54649, 8624, 826, 44, 1, 3628800, 6999840, 5753736, 2655764
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| Sum of entries in row n = n!/2 = A001710(n). T(n,1)=(n-1)!=A000142(n-1). Columns 2,3,4 and 5 yield A001705,A001706,A001707 and A001708, respectively.
See A143491 for the interpretation of these numbers as restricted Stirling numbers of the first kind. See A049444 for a signed version of this array. [From Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008]
With offset n=0,k=0 : triangle T(n,k), read by rows, given by [2,1,3,2,4,3,5,4,6,5,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. [From DELEHAM Philippe, Sep 29 2011]
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FORMULA
| E.g.f.=Sum[(1/n!)T(n,k)x^n*t^k, k=1..n-1, n>=2]=1/[(1+t)(1-x)^t]-(1+tx)/(1+t). Generating polynomial of row n = t*Product(j+t, j=2..n-1). T(n,k) is the sum of all products of n-k-1 different integers taken from {2,3,...,n-1}. For example, T(6,3)=2*3+2*4+2*5+3*4+3*5+4*5=71.
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EXAMPLE
| T(6,3)=71 because (-1)^9*[s(6,1)+s(6,2)+s(6,3)]=-(-120+274-225)=71.
Triangle starts:
1;
2,1;
6,5,1;
24,26,9,1;
120,154,71,14,1;
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MAPLE
| A136124_row := proc(n) local k, j; `if`(n=0, 1, seq((-1)^(n+1-k)*add(stirling1(n+1, j), j=1..k), k=1..n)) end: seq(print(A136124_row(r)), r=1..6); # Peter Luschny, Sep. 29 2011
with(combinat): T:=proc(n, k) options operator, arrow: (-1)^(n+k)*(sum(stirling1(n, j), j=1..k)) end proc: for n from 2 to 11 do seq(T(n, k), k=1..n-1) end do; # yields sequence in triangular form
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CROSSREFS
| Cf. A000142, A008275, A001705, A001706, A001707, A001708, A001710.
Cf. A049444, A143491. [From Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008]
Sequence in context: A121575 A049444 * A143491 A070918 A113381 A118980
Adjacent sequences: A136121 A136122 A136123 * A136125 A136126 A136127
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 23 2007
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