The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2). 16
 1, 0, 1, 1, -2, 1, 0, 9, -6, 1, 9, -36, 42, -12, 1, 0, 225, -300, 130, -20, 1, 225, -1350, 2475, -1380, 315, -30, 1, 0, 11025, -22050, 15435, -4620, 651, -42, 1, 11025, -88200, 220500, -182280, 67830, -12600, 1204, -56, 1, 0, 893025, -2381400, 2302020, -1020600, 235494, -29736, 2052, -72 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS This is a Sheffer triangle (lower triangular exponential convolution matrix). For Sheffer row polynomials see the S. Roman reference and explanations under A048854. In the umbral notation of the S. Roman reference this would be called Sheffer for ((sqrt(1-2*t))/(1-t), t/(1-t)). The associated Sheffer triangle is A111596. Matrix logarithm equals A112239. - Paul D. Hanna, Aug 29 2005 The row polynomials (1/2^n)* H(n,sqrt(x/2))^2, with the Hermite polynomials H(n,x), have e.g.f. (1/sqrt(1-y^2))*exp(x*y/(1+y)). The row polynomials s(n,x):=sum(a(n,m)*x^m,m=0..n), together with the associated row polynomials p(n,x) of A111596, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0. The unsigned column sequences are: A111601, A111602, A111777-A111784, for m=1..10. REFERENCES R. P. Boas and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer, 1958, p. 41 S. Roman, The Umbral Calculus, Academic Press, New York, 1984, p. 128. LINKS G. C. Greubel, Rows n=0..100 of triangle, flattened FORMULA E.g.f. for column m>=0: (1/sqrt(1-x^2))*((x/(1+x))^m)/m!. a(n, m)=((-1)^(n-m))*(n!/m!)*sum(binomial(2*k, k)*binomial(n-2*k-1, m-1)/(4^k), k=0..floor((n-m)/2)), n>=m>=1. a(2*k, 0)= ((2*k)!/(k!*2^k))^2 = A001818(k), a(2*k+1) = 0, k>=0. a(n, m)=0 if n

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 23 07:28 EST 2020. Contains 332159 sequences. (Running on oeis4.)