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A048854
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Triangle of coefficients of certain Sheffer-polynomials.
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25
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1, 2, 1, 12, 12, 1, 120, 180, 30, 1, 1680, 3360, 840, 56, 1, 30240, 75600, 25200, 2520, 90, 1, 665280, 1995840, 831600, 110880, 5940, 132, 1, 17297280, 60540480, 30270240, 5045040, 360360, 12012, 182, 1, 518918400, 2075673600, 1210809600
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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(1/sqrt(1+4*t),t/(1+4*t)).
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REFERENCES
| S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
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FORMULA
| a(n, m) = (n!/m!)*A046521(n, m) = (n!/m!)* binomial(2*n, n)*binomial(n, m)/binomial(2*m, m), n >= m >= 0, a(n, m) := 0, n<m.
Sum_{n>=0, k>=0} a(n, k)*x^n*y^k/(2*n)! = exp(x)*cosh(sqrt(x*y)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 21 2003
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EXAMPLE
| 1;
2,1;
12,12,1;
120,180,30,1;
1680,3360,840,56,1;
30240,75600,25200,2520,90,1;
665280,1995840,831600,110880,5940,132,1;
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CROSSREFS
| Related to triangle A046521. Cf. A048786. a(n, 0) = A001813.
Sequence in context: A135256 A090586 A199930 * A151508 A164826 A055392
Adjacent sequences: A048851 A048852 A048853 * A048855 A048856 A048857
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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