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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. 8
 1, 1, 1, 2, 1, 6, 1, 12, 12, 1, 20, 60, 1, 30, 180, 120, 1, 42, 420, 840, 1, 56, 840, 3360, 1680, 1, 72, 1512, 10080, 15120, 1, 90, 2520, 25200, 75600, 30240, 1, 110, 3960, 55440, 277200, 332640, 1, 132, 5940, 110880, 831600, 1995840, 665280, 1, 156 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 REFERENCES 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. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50. LINKS G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018. FORMULA E.g.f.: exp(x^2+y*x). - Vladeta Jovovic, Feb 21 2003 a(n, k) = n!/(k! (n-2k)!). - Dean Hickerson, Feb 24 2003 EXAMPLE Triangle begins   1;   1;   1,     2;   1,     6;   1,    12,    12;   1,    20,    60;   1,    30,   180,   120;   1,    42,   420,   840;   1,    56,   840,  3360,  1680;   1,    72,  1512, 10080, 15120; 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 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). MATHEMATICA Flatten[Table[n!/(k! * (n-2k)!), {n, 0, 13}, {k, 0, Floor[n/2]}]] (* Second program: *) 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 *) PROG (PARI) for(n=0, 25, for(k=0, floor(n/2), print1(n!/(k!*(n-2*k)!), ", "))) \\ G. C. Greubel, Jan 07 2017 CROSSREFS Cf. A059343, A060821. Cf. A119275 (signed row reverse). Sequence in context: A233809 A207536 A060173 * A109193 A225769 A280736 Adjacent sequences:  A059341 A059342 A059343 * A059345 A059346 A059347 KEYWORD nonn,easy,nice,tabf AUTHOR N. J. A. Sloane, Jan 27 2001 EXTENSIONS More terms from Vladeta Jovovic, Feb 21 2003 Edited by Emeric Deutsch, Jun 05 2004 STATUS approved

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Last modified November 16 12:40 EST 2018. Contains 317272 sequences. (Running on oeis4.)