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

0,4

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 A322672 A363595

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