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