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A129062
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Matrix product of Stirling2 with unsigned Stirling1 triangle.
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3
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1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 26, 36, 12, 1, 0, 150, 250, 120, 20, 1, 0, 1082, 2040, 1230, 300, 30, 1, 0, 9366, 19334, 13650, 4270, 630, 42, 1, 0, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 0, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| For the subtriangle without column nr. m=0 and row nr. n=0 see A079641.
The reversed matrix product |S1|. S2 is given in A111596.
As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
E.g.f. for row polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n) is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
A048993*A132393 as infinite lower triangular matrices. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 01 2009]
Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) whre DELTA is the operator defined in A084938. - From DELEHAM Philippe, Nov 19 2011.
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LINKS
| W. Lang, First ten rows and more.
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FORMULA
| a(n,m)=sum(S2(n,k)*|S1(k,m)|,k=m..n), n>=0. S2(n,m):=A048993. S1(n,m):=A048994.
E.g.f. column nr. m (with leading zeros): (f(x)^m)/m! with f(x):= -ln(1-(exp(x)-1))= -ln(2-exp(x)).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - From DELEHAM Philippe, Nov 19 2011.
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EXAMPLE
| Rows :[1]; [0,1]; [0,2,1]; [0,6,6,1]; [0,26,36,12,1]; [0,150,250,120,20,1]; ...
Triangle begins :
1
0, 1
0, 2, 1
0, 6, 6, 1
0, 26, 36, 12, 1
0, 150, 250, 120, 20, 1
0, 1082, 2040, 1230, 300, 30, 1,
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CROSSREFS
| Cf. A000670,A005649
Sequence in context: A021830 A111184 A111596 * A163936 A187555 A117651
Adjacent sequences: A129059 A129060 A129061 * A129063 A129064 A129065
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KEYWORD
| nonn,tabl,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) May 04 2007
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