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Search: id:A075502
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| A075502 |
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Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7). |
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+0
10
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| 1, 7, 1, 49, 21, 1, 343, 343, 42, 1, 2401, 5145, 1225, 70, 1, 16807, 74431, 30870, 3185, 105, 1, 117649, 1058841, 722701, 120050, 6860, 147, 1, 823543, 14941423, 16235562, 4084101, 360150, 13034, 196, 1
(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 D. E. 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(7*z)-1)*x/7)-1.
Row sums give A075506(n),n>=1. The columns (without leading zeros) give A000420 (powers of 7), A075921-A075925, A076002 for m=1..7.
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FORMULA
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a(n, m)=(7^(n-m))*S2(n, m) with S2(n, m) := A008277(n, m) (Stirling2).
a(n, m)=7*m*a(n-1, m) + a(n-1, m-1), n>=m>=1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
a(n, m)=sum(A075513(m, p)*((p+1)*7)^(n-m), p=0..m-1)/(m-1)! for n>=m>=1 else 0.
G.f. for m-th column: (x^m)/product(1-7*k*x, k=1..m), m>=1.
E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m>=1.
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EXAMPLE
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[1]; [7,1]; [49,21,1]; ...; p(3,x)=x*(49+21*x+x^2).
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CROSSREFS
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Cf. A075501, A075503.
Sequence in context: A051931 A038267 A027466 this_sequence A052104 A144450 A051339
Adjacent sequences: A075499 A075500 A075501 this_sequence A075503 A075504 A075505
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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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