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
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