cp's OEIS Frontend

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.

%I A220178 #20 Jun 22 2019 09:22:01
%S A220178 1,1,2,3,6,6,7,24,30,20,19,80,150,140,70,51,270,630,840,630,252,141,
%T A220178 882,2520,4200,4410,2772,924,393,2856,9576,19320,25410,22176,12012,
%U A220178 3432,1107,9144,35280,83160,131670,144144,108108,51480,12870,3139,29070,126720,341880,630630,828828,780780,514800,218790,48620
%N 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.
%H A220178 Ivan Neretin, <a href="/A220178/b220178.txt">Table of n, a(n) for n = 0..5150 (rows 0..100)</a>
%F A220178 G.f.: A(x,y) = 1 / sqrt(1-2*x-3*x^2 - 4*x*y).
%F A220178 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).
%F A220178 First column is the central trinomial coefficients (A002426).
%F A220178 Main diagonal is the central binomial coefficients (A000984).
%F A220178 Row sums form the central coefficients of (1+3*x+3*x^2)^n (A122868).
%e A220178 Triangle begins:
%e A220178 1;
%e A220178 1, 2;
%e A220178 3, 6, 6;
%e A220178 7, 24, 30, 20;
%e A220178 19, 80, 150, 140, 70;
%e A220178 51, 270, 630, 840, 630, 252;
%e A220178 141, 882, 2520, 4200, 4410, 2772, 924;
%e A220178 393, 2856, 9576, 19320, 25410, 22176, 12012, 3432;
%e A220178 1107, 9144, 35280, 83160, 131670, 144144, 108108, 51480, 12870; ...
%e A220178 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.
%e A220178 The g.f. for row n is d^n/dx^n (1+x+x^2)^n/n!, which begins:
%e A220178 n=0: 1;
%e A220178 n=1: 1 + 2*x;
%e A220178 n=2: 3 + 6*x + 6*x^2;
%e A220178 n=3: 7 + 24*x + 30*x^2 + 20*x^3;
%e A220178 n=4: 19 + 80*x + 150*x^2 + 140*x^3 + 70*x^4;
%e A220178 n=5: 51 + 270*x + 630*x^2 + 840*x^3 + 630*x^4 + 252*x^5;
%e A220178 n=6: 141 + 882*x + 2520*x^2 + 4200*x^3 + 4410*x^4 + 2772*x^5 + 924*x^6; ...
%t A220178 Flatten@Table[CoefficientList[D[(1 + x + x^2)^n/n!, {x, n}], x], {n, 0, 9}] (* _Ivan Neretin_, Jun 22 2019 *)
%o A220178 (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)}
%o A220178 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%o A220178 (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
%Y A220178 Cf. A002426 (first column), A000984 (main diagonal), A122868 (row sums).
%K A220178 nonn,tabl
%O A220178 0,3
%A A220178 _Paul D. Hanna_, Dec 06 2012