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A059344 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.

%I A059344 #37 Jul 11 2018 20:11:52
%S A059344 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 A059344 1680,1,72,1512,10080,15120,1,90,2520,25200,75600,30240,1,110,3960,
%U A059344 55440,277200,332640,1,132,5940,110880,831600,1995840,665280,1,156
%N A059344 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 A059344 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 A059344 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.
%H A059344 G. C. Greubel, <a href="/A059344/b059344.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H A059344 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 A059344 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 A059344 E.g.f.: exp(x^2+y*x). - _Vladeta Jovovic_, Feb 21 2003
%F A059344 a(n, k) = n!/(k! (n-2k)!). - _Dean Hickerson_, Feb 24 2003
%e A059344 Triangle begins
%e A059344   1;
%e A059344   1;
%e A059344   1,     2;
%e A059344   1,     6;
%e A059344   1,    12,    12;
%e A059344   1,    20,    60;
%e A059344   1,    30,   180,   120;
%e A059344   1,    42,   420,   840;
%e A059344   1,    56,   840,  3360,  1680;
%e A059344   1,    72,  1512, 10080, 15120;
%e A059344 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 A059344 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 A059344 Flatten[Table[n!/(k! * (n-2k)!), {n, 0, 13}, {k, 0, Floor[n/2]}]]
%t A059344 (* Second program: *)
%t A059344 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 A059344 (PARI) for(n=0,25, for(k=0,floor(n/2), print1(n!/(k!*(n-2*k)!), ", "))) \\ _G. C. Greubel_, Jan 07 2017
%Y A059344 Cf. A059343, A060821.
%Y A059344 Cf. A119275 (signed row reverse).
%K A059344 nonn,easy,nice,tabf
%O A059344 0,4
%A A059344 _N. J. A. Sloane_, Jan 27 2001
%E A059344 More terms from _Vladeta Jovovic_, Feb 21 2003
%E A059344 Edited by _Emeric Deutsch_, Jun 05 2004