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%I #37 Jul 11 2018 20:11:52
%S 1,1,1,2,1,6,1,12,12,1,20,60,1,30,180,120,1,42,420,840,1,56,840,3360,
%T 1680,1,72,1512,10080,15120,1,90,2520,25200,75600,30240,1,110,3960,
%U 55440,277200,332640,1,132,5940,110880,831600,1995840,665280,1,156
%N Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.
%H G. C. Greubel, <a href="/A059344/b059344.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.
%F E.g.f.: exp(x^2+y*x). - _Vladeta Jovovic_, Feb 21 2003
%F a(n, k) = n!/(k! (n-2k)!). - _Dean Hickerson_, Feb 24 2003
%e Triangle begins
%e 1;
%e 1;
%e 1, 2;
%e 1, 6;
%e 1, 12, 12;
%e 1, 20, 60;
%e 1, 30, 180, 120;
%e 1, 42, 420, 840;
%e 1, 56, 840, 3360, 1680;
%e 1, 72, 1512, 10080, 15120;
%e x^2 = 1/2^2*(Hermite(2,x)+2*Hermite(0,x)); x^3 = 1/2^3*(Hermite(3,x)+6*Hermite(1,x)); x^4 = 1/2^4*(Hermite(4,x)+12*Hermite(2,x)+12*Hermite(0,x)); x^5 = 1/2^5*(Hermite(5,x)+20*Hermite(3,x)+60*Hermite(1,x)); x^6 = 1/2^6*(Hermite(6,x)+30*Hermite(4,x)+180*Hermite(2,x)+120*Hermite(0,x)). - _Vladeta Jovovic_, Feb 21 2003
%e 1 = H(0); 2x = H(1); 4x^2 = H(2)+2H(0); 8x^3 = H(3)+6H(1); etc. where H(k)=Hermite(k,x).
%t Flatten[Table[n!/(k! * (n-2k)!), {n, 0, 13}, {k, 0, Floor[n/2]}]]
%t (* Second program: *)
%t row[n_] := Table[h[k], {k, n, Mod[n, 2], -2}] /. SolveAlways[2^n*x^n == Sum[h[k]*HermiteH[k, x], {k, Mod[n, 2], n, 2}], x] // First; Table[ row[n], {n, 0, 13}] // Flatten (* _Jean-François Alcover_, Jan 05 2016 *)
%o (PARI) for(n=0,25, for(k=0,floor(n/2), print1(n!/(k!*(n-2*k)!), ", "))) \\ _G. C. Greubel_, Jan 07 2017
%Y Cf. A059343, A060821.
%Y Cf. A119275 (signed row reverse).
%K nonn,easy,nice,tabf
%O 0,4
%A _N. J. A. Sloane_, Jan 27 2001
%E More terms from _Vladeta Jovovic_, Feb 21 2003
%E Edited by _Emeric Deutsch_, Jun 05 2004
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