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.

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

Offset: 0

Author

N. J. A. Sloane, Jan 27 2001

Keywords

• nonn
• easy
• nice
• tabf

Examples

			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).
		

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.

Crossrefs

Cf. A059343, A060821.

Cf. A119275 (signed row reverse).

Programs

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 *)

PARI

for(n=0,25, for(k=0,floor(n/2), print1(n!/(k!*(n-2*k)!), ", "))) \\ G. C. Greubel, Jan 07 2017

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

Extensions

More terms from Vladeta Jovovic, Feb 21 2003

Edited by Emeric Deutsch, Jun 05 2004