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Search: id:A075500
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| A075500 |
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Stirling2 triangle with scaled diagonals (powers of 5). |
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+0
10
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| 1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := sum(a(n,m)x^m,m=1..n), n>=1, have e.g.f. J(x; z)= exp((exp(5*z)-1)*x/5)-1.
Row sums give A005011(n-1),n>=1. The columns (without leading zeros) give A000351 (powers of 5), A016164, A075911-A075915 for m=1..7.
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FORMULA
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a(n, m)=(5^(n-m))S2(n, m) with S2(n, m) := A008277(n, m) (Stirling2).
a(n, m)=sum((A075513(m, p)*((p+1)*5)^(n-m))/(m-1)!, p=0..m-1) for n>=m>=1 else 0.
a(n, m)=5m*a(n-1, m) + a(n-1, m-1), n>=m>=1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/product(1-5k*x, k=1..m), m>=1.
E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m>=1.
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EXAMPLE
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[1];[5,1];[25,15,1]; ...; p(3,x)=x(25+15*x+x^2).
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CROSSREFS
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Cf. A075499, A075501.
Sequence in context: A162259 A077195 A038243 this_sequence A096645 A140713 A125906
Adjacent sequences: A075497 A075498 A075499 this_sequence A075501 A075502 A075503
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002
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