OFFSET
0,3
LINKS
Ivan Neretin, Table of n, a(n) for n = 0..5150 (rows 0..100)
FORMULA
G.f.: A(x,y) = 1 / sqrt(1-2*x-3*x^2 - 4*x*y).
G.f.: A(x,y) = Sum_{k>=0} binomial(2*k,k) * x^k*y^k / (1-2*x-3*x^2)^(k+1/2).
First column is the central trinomial coefficients (A002426).
Main diagonal is the central binomial coefficients (A000984).
Row sums form the central coefficients of (1+3*x+3*x^2)^n (A122868).
EXAMPLE
Triangle begins:
1;
1, 2;
3, 6, 6;
7, 24, 30, 20;
19, 80, 150, 140, 70;
51, 270, 630, 840, 630, 252;
141, 882, 2520, 4200, 4410, 2772, 924;
393, 2856, 9576, 19320, 25410, 22176, 12012, 3432;
1107, 9144, 35280, 83160, 131670, 144144, 108108, 51480, 12870; ...
The g.f. for column k>=0 equals the central binomial coefficient C(2*k,k) times x^k*y^k*G(x)^(2*k+1) where G(x) = 1/sqrt(1-2*x-3*x^2) is the g.f. of the central trinomial coefficients A002426.
The g.f. for row n is d^n/dx^n (1+x+x^2)^n/n!, which begins:
n=0: 1;
n=1: 1 + 2*x;
n=2: 3 + 6*x + 6*x^2;
n=3: 7 + 24*x + 30*x^2 + 20*x^3;
n=4: 19 + 80*x + 150*x^2 + 140*x^3 + 70*x^4;
n=5: 51 + 270*x + 630*x^2 + 840*x^3 + 630*x^4 + 252*x^5;
n=6: 141 + 882*x + 2520*x^2 + 4200*x^3 + 4410*x^4 + 2772*x^5 + 924*x^6; ...
MATHEMATICA
Flatten@Table[CoefficientList[D[(1 + x + x^2)^n/n!, {x, n}], x], {n, 0, 9}] (* Ivan Neretin, Jun 22 2019 *)
PROG
(PARI) {T(n, k)=polcoeff(polcoeff(1/sqrt(1-2*x-3*x^2 - 4*x*y +x*O(x^n)+y*O(y^k)), n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) row(n) = my(p=(1+x+x^2)^n / n!); for (k=1, n, p = deriv(p)); Vecrev(p); \\ Michel Marcus, Jun 22 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 06 2012
STATUS
approved