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A039761
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Triangle of D-analogs of Stirling numbers of 2nd kind.
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1
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1, 1, 0, 1, 2, 1, 1, 6, 7, 1, 1, 12, 34, 24, 1, 1, 20, 110, 190, 81, 1, 1, 30, 275, 920, 1051, 268, 1, 1, 42, 581, 3255, 7371, 5747, 869, 1, 1, 56, 1092, 9296, 35686, 57568, 31060, 2768, 1, 1, 72, 1884, 22764, 134022, 373926, 441652, 166068, 8689, 1, 1, 90, 3045, 49680, 418362, 1812552, 3803290, 3342240, 879541, 26964, 1
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OFFSET
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0,5
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COMMENTS
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Since T(n,k) = A039760(n,n-k), we have Sum_{n,k >= 0} T(n,k)*(x^n/n!)*y^k = Sum_{n,k >= 0} A039760(n,n-k)*((x*y)^n/n!)*(1/y)^(n-k) = Sum(n,m >= 0} A039760(n,m)*((x*y)^n/n!)*(1/y)^m. Thus, to get the bivariate e.g.f.-o.g.f. of T(n,k), we perform the following transformation in the bivariate e.g.f.-o.g.f. of A039760: (x,y) -> (x*y, 1/y). - Petros Hadjicostas, Jul 11 2020
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LINKS
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FORMULA
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Bivariate e.g.f.-o.g.f.: (exp(x*y) - x*y) * exp(1/(2*y)*(exp(2*x*y) - 1)). [Apply (x, y) -> (x*y, 1/y) to (exp(x) - x)*exp(y/2*(exp(2*x) - 1)). - Petros Hadjicostas, Jul 11 2020]
T(n,k) = (Sum_{j=n-k..n} 2^(j+k-n)*binomial(n,j)*Stirling2(j, n-k)) - 2^(k-1)*n*Stirling2(n-1, n-k). [Use Proposition 3 in Suter (2000) with k -> n-k.] - Petros Hadjicostas, Jul 11 2020
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EXAMPLE
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Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
1, 0;
1, 2, 1;
1, 6, 7, 1;
1, 12, 34, 24, 1;
1, 20, 110, 190, 81, 1;
1, 30, 275, 920, 1051, 268, 1;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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Ruedi Suter (suter(AT)math.ethz.ch)
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EXTENSIONS
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STATUS
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approved
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