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A075501
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Stirling2 triangle with scaled diagonals (powers of 6).
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8
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1, 6, 1, 36, 18, 1, 216, 252, 36, 1, 1296, 3240, 900, 60, 1, 7776, 40176, 19440, 2340, 90, 1, 46656, 489888, 390096, 75600, 5040, 126, 1, 279936, 5925312, 7511616, 2204496, 226800, 9576, 168, 1, 1679616, 71383680
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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(6*z)-1)*x/6)-1.
Row sums give A005012(n-1),n>=1. The columns (without leading zeros) give A000400 (powers of 6), A016175, A075916-A075920 for m=1..7.
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FORMULA
| a(n, m)=(6^(n-m))S2(n, m) with S2(n, m) := A008277(n, m) (Stirling2).
a(n, m)=sum((A075513(m, p)*((p+1)*6)^(n-m))/(m-1)!, p=0..m-1) for n>=m>=1 else 0.
a(n, m)=6m*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(1-6k*x, k=1..m), m>=1.
E.g.f. for m-th column: (((exp(6x)-1)/6)^m)/m!, m>=1.
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EXAMPLE
| [1];[6,1];[36,18,1]; ...; p(3,x)=x(36+18*x+x^2).
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CROSSREFS
| Cf. A075500, A075502.
Sequence in context: A051930 A147320 A038255 * A089504 A145927 A113365
Adjacent sequences: A075498 A075499 A075500 * A075502 A075503 A075504
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002
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