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Search: id:A048870
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| A048870 |
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Triangle of coefficients of certain Sheffer-polynomials. |
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1
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| 1, 1, 1, 4, 10, 1, 30, 132, 27, 1, 336, 2232, 696, 52, 1, 5040, 46320, 19500, 2200, 85, 1, 95040, 1141920, 606960, 91800, 5340, 126, 1, 2162160, 32639040, 20991600, 3986640, 310170, 11004, 175, 1, 57657600, 1061746560, 802287360, 183550080
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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s(n,x) := sum(a(n,m)*x^m,m=0..n) are monic polynomials satisfying s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(A048786(n,m)*x^m, m=1..n) (row polynomials of triangle A048786) and p(0,x)=1. In the umbral calculus (see reference) the s(n,x) are called Sheffer polynomials for(c(t/(1+4*t)),t/(1+4*t)), where c(x) = g.f. for Catalan numbers A000108. a(n,0) = A001761(n-2) = n!*A000108(n).
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REFERENCES
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S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
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FORMULA
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a(n, m) = (n!/m!)*A046527(n, m) = (n!/m!)*binomial(n, m-1)*(4^(n-m+1)-binomial(2*n, n)/binomial(2*(m-1), m-1))/2, n >= m >= 0, a(n, m) := 0, n<m.
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CROSSREFS
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A046527, A048786, A000108, A001761.
Sequence in context: A008345 A016488 A087212 this_sequence A070261 A054048 A121512
Adjacent sequences: A048867 A048868 A048869 this_sequence A048871 A048872 A048873
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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