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A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

%I A111595 #27 Mar 12 2021 15:57:30
%S A111595 1,0,1,1,-2,1,0,9,-6,1,9,-36,42,-12,1,0,225,-300,130,-20,1,225,-1350,
%T A111595 2475,-1380,315,-30,1,0,11025,-22050,15435,-4620,651,-42,1,11025,
%U A111595 -88200,220500,-182280,67830,-12600,1204,-56,1,0,893025,-2381400,2302020,-1020600,235494,-29736,2052,-72
%N A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).
%C A111595 This is a Sheffer triangle (lower triangular exponential convolution matrix). For Sheffer row polynomials see the S. Roman reference and explanations under A048854.
%C A111595 In the umbral notation of the S. Roman reference this would be called Sheffer for ((sqrt(1-2*t))/(1-t), t/(1-t)).
%C A111595 The associated Sheffer triangle is A111596.
%C A111595 Matrix logarithm equals A112239. - _Paul D. Hanna_, Aug 29 2005
%C A111595 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)).
%C A111595 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.
%C A111595 The unsigned column sequences are: A111601, A111602, A111777-A111784, for m=1..10.
%D A111595 R. P. Boas and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer, 1958, p. 41
%D A111595 S. Roman, The Umbral Calculus, Academic Press, New York, 1984, p. 128.
%H A111595 G. C. Greubel, <a href="/A111595/b111595.txt">Rows n=0..100 of triangle, flattened</a>
%F A111595 E.g.f. for column m>=0: (1/sqrt(1-x^2))*((x/(1+x))^m)/m!.
%F A111595 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.
%e A111595 The triangle a(n, m) begins:
%e A111595 n\m       0         1         2          3         4         5       6       7     8    9  10 ...
%e A111595 0:        1
%e A111595 1:        0         1
%e A111595 2:        1        -2         1
%e A111595 3:        0         9        -6          1
%e A111595 4:        9       -36        42        -12         1
%e A111595 5:        0       225      -300        130       -20         1
%e A111595 6:      225     -1350      2475      -1380       315       -30       1
%e A111595 7:        0     11025    -22050      15435     -4620       651     -42       1
%e A111595 8:    11025    -88200    220500    -182280     67830    -12600    1204     -56     1
%e A111595 9:        0    893025  -2381400    2302020  -1020600    235494  -29736    2052   -72    1
%e A111595 10:  893025  -8930250  28279125  -30958200  15961050  -4396140  689850  -63000  3285  -90   1
%e A111595 -------------------------------------------------------------------------------------------------
%t A111595 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 *)
%o A111595 (Python)
%o A111595 from sympy import hermite, Poly, sqrt, symbols
%o A111595 x = symbols('x')
%o A111595 def a(n): return Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()[::-1]
%o A111595 for n in range(11): print(a(n)) # _Indranil Ghosh_, May 26 2017
%Y A111595 Row sums: A111882. Unsigned row sums: A111883.
%Y A111595 Cf. A112239 (matrix log).
%K A111595 sign,easy,tabl
%O A111595 0,5
%A A111595 _Wolfdieter Lang_, Aug 23 2005