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.
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<m.
EXAMPLE
The triangle a(n, m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 0 1
2: 1 -2 1
3: 0 9 -6 1
4: 9 -36 42 -12 1
5: 0 225 -300 130 -20 1
6: 225 -1350 2475 -1380 315 -30 1
7: 0 11025 -22050 15435 -4620 651 -42 1
8: 11025 -88200 220500 -182280 67830 -12600 1204 -56 1
9: 0 893025 -2381400 2302020 -1020600 235494 -29736 2052 -72 1
10: 893025 -8930250 28279125 -30958200 15961050 -4396140 689850 -63000 3285 -90 1
-------------------------------------------------------------------------------------------------
MATHEMATICA
row[n_] := CoefficientList[ 1/2^n*HermiteH[n, Sqrt[x/2]]^2, x]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)
PROG
(Python)
from sympy import hermite, Poly, sqrt, symbols
x = symbols('x')
def a(n): return Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()[::-1]
for n in range(11): print(a(n)) # Indranil Ghosh, May 26 2017
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Aug 23 2005
STATUS
approved