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A145324
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Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
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7
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1, 1, 2, 1, 5, 6, 1, 9, 26, 24, 1, 14, 71, 154, 120, 1, 20, 155, 580, 1044, 720, 1, 27, 295, 1665, 5104, 8028, 5040, 1, 35, 511, 4025, 18424, 48860, 69264, 40320, 1, 44, 826, 8624, 54649, 214676, 509004, 663696, 362880, 1, 54, 1266, 16884, 140889, 761166
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| The last number of row n is n!
Essentially the triangle given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] where DELTA is the operator defined in A084938. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 09 2008]
T(n+1,k+1) = a_k(2,3,...,n+1), n>=0, k=0..n, with the elementary symmetric function a_k(x[1],x[2],...,x[n]),
with a_0(0):=1. E.g., a_2(2,3,4)= 2*3+2*4+3*4=26 =
T(4,3). - From Wolfdieter Lang, Oct 24 2011.
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FORMULA
| T(n,k) = A143491(n+1,n+2-k). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 10 2008]
T(n,k) = sum((-1)^m*|s(n+1,n+2-k+m)|,m=0..k-1), n>=1, k=1..n, with the Stirling numbers of the first kind s(n,k)=A048994(n,k). - From Wolfdieter Lang, Oct 24 2011.
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EXAMPLE
| From Wolfdieter Lang, Oct 24 2011 (Start)
n\k 1 2 3 4 5 6 7 ...
1: 1
2: 1 2
3: 1 5 6
4: 1 9 26 24
5: 1 14 71 154 120
6: 1 20 155 580 1044 720
7: 1 27 295 1665 5104 8028 5040
...
T(4,3)= 26 = |s(5,3)| - |s(5,4)| + |s(5,5)| = 35-10+1.
(End)
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MAPLE
| A145324 := proc(n, k) coeftayl( 1*mul(x+i, i=2..n), x=0, n-k) ; end: for n from 1 to 11 do for k from 1 to n do printf("%d, ", A145324(n, k)) ; od: od: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 10 2008]
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CROSSREFS
| Sequence in context: A193816 A193722 A193635 * A179457 A107783 A047887
Adjacent sequences: A145321 A145322 A145323 * A145325 A145326 A145327
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KEYWORD
| nonn,tabl
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AUTHOR
| Jose Ramon Real (joseramonreal(AT)yahoo.es), Oct 07 2008
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 10 2008
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