A220178 Triangle where the g.f. for row n equals d^n/dx^n (1+x+x^2)^n / n! for n>=0, as read by rows.

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, 3139, 29070, 126720, 341880, 630630, 828828, 780780, 514800, 218790, 48620

Offset: 0

Author

Paul D. Hanna, Dec 06 2012

Keywords

• nonn
• tabl

Examples

			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; ...
		

Links

• Ivan Neretin, Table of n, a(n) for n = 0..5150 (rows 0..100)

Crossrefs

Cf. A002426 (first column), A000984 (main diagonal), A122868 (row sums).

Programs

Mathematica

Flatten@Table[CoefficientList[D[(1 + x + x^2)^n/n!, {x, n}], x], {n, 0, 9}] (* Ivan Neretin, Jun 22 2019 *)

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

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).