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Search: id:A132393
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| A132393 |
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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 . |
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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
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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A094645*A007318 as infinite lower triangular matrices . Unsigned Stirling numbers of first kind A048994 .
Row sums are the factorial numbers. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 18 2008
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REFERENCES
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Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150
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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), A113136(n), A113137(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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 (rlbagulatftn(AT)yahoo.com), Apr 18 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 ;...
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MATHEMATICA
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Clear[p, g] 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 (rlbagulatftn(AT)yahoo.com), Apr 18 2008
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CROSSREFS
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Cf. A008275, A048994, A130534.
Sequence in context: A081247 A005210 A048994 this_sequence A121434 A137329 A004579
Adjacent sequences: A132390 A132391 A132392 this_sequence 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 (kolotoko(AT)wanadoo.fr), Nov 10 2007
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