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Search: id:A075497
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| A075497 |
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Stirling2 triangle with scaled diagonals (powers of 2). |
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+0
10
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| 1, 2, 1, 4, 6, 1, 8, 28, 12, 1, 16, 120, 100, 20, 1, 32, 496, 720, 260, 30, 1, 64, 2016, 4816, 2800, 560, 42, 1, 128, 8128, 30912, 27216, 8400, 1064, 56, 1, 256, 32640, 193600, 248640, 111216, 21168, 1848, 72
(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(2*z)-1)*x/2)-1.
Row sums give A004211(n),n>=1. The columns (without leading zeros) give A000079 (powers of 2), A006516, A016283, A025966, A075510-2 for m=1..7.
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LINKS
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W. Lang, First 10 rows .
T. Mansour, Generalization of some identities involving the Fibonacci numbers
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FORMULA
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a(n, m)=(2^(n-m))*S2(n, m) with S2(n, m) := A008277(n, m) (Stirling2).
a(n, m)=sum(A075513(m, p)*((p+1)*2)^(n-m), p=0..m-1)/(m-1)! for n>=m>=1 else 0.
a(n, m)=2*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.
G.f. for m-th column: (x^m)/product(1-2*k*x, k=1..m), m>=1.
E.g.f. for m-th column: (((exp(2*x)-1)/2)^m)/m!, m>=1.
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EXAMPLE
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[1];[2,1];[4,6,1]; ...; p(3,x)=x*(4+6*x+x^2).
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CROSSREFS
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Cf. A008277, A075498.
Sequence in context: A109822 A114192 A114656 this_sequence A158983 A079474 A091543
Adjacent sequences: A075494 A075495 A075496 this_sequence A075498 A075499 A075500
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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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