A048870 Triangle of coefficients of certain Sheffer-polynomials.

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

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• tabl

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(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).

References

• S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

Crossrefs

A046527, A048786, A000108, A001761.

Formula

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

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