A322116 Main diagonal of triangle A321600; a(n) = A321600(n,n-1) for n >= 1.

2, 6, 26, 78, 242, 726, 2186, 6558, 19682, 59046, 177146, 531438, 1594322, 4782966, 14348906, 43046718, 129140162, 387420486, 1162261466, 3486784398, 10460353202, 31381059606, 94143178826, 282429536478, 847288609442, 2541865828326, 7625597484986, 22876792454958, 68630377364882, 205891132094646, 617673396283946, 1853020188851838, 5559060566555522, 16677181699666566

Offset: 1

Author

Paul D. Hanna, Nov 26 2018

Keywords

• nonn

Comments

Triangle A321600 describes log( (1-y)*Sum_{n=-oo...+oo} (x^n + y)^n )/(1-y).

Examples

			G.f.: A(x) = 2*x + 6*x^2 + 26*x^3 + 78*x^4 + 242*x^5 + 726*x^6 + 2186*x^7 + 6558*x^8 + 19682*x^9 + 59046*x^10 + ...
L.g.f.: L(x)  =  log( (1-x)*(1-x^2)/(1-3*x) )  =  2*x + 6*x^2/2 + 26*x^3/3 + 78*x^4/4 + 242*x^5/5 + 726*x^6/6 + 2186*x^7/7 + 6558*x^8/8 + 19682*x^9/9 + 59046*x^10/10 + 177146*x^11/11 + ... + A321600(n,n-1)*x^n/n + ...
such that
exp(L(x)) = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 144*x^5 + 432*x^6 + 1296*x^7 + 3888*x^8 + 11664*x^9 + 34992*x^10 + 104976*x^11 + ... + A257970(n)*x^n + ...
exp(L(x)/2) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 44*x^5 + 122*x^6 + 342*x^7 + 966*x^8 + 2746*x^9 + 7846*x^10 + 22514*x^11 + 64836*x^12 + ... + A105696(n)*x^n + ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3, 1, -3).

Crossrefs

Cf. A321600, A257970, A105696.

Programs

PARI

{a(n) = n*polcoeff( log((1 - x)*(1 - x^2)/(1 - 3*x +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))

Formula

L.g.f.: log( (1 - x)*(1 - x^2)/(1 - 3*x) ).

G.f.: 2*x*(1 + 3*x^2)/((1 - x^2)*(1 - 3*x)).