A322116 - OEIS

%I #7 Jul 04 2023 14:08:04

%S 2,6,26,78,242,726,2186,6558,19682,59046,177146,531438,1594322,

%T 4782966,14348906,43046718,129140162,387420486,1162261466,3486784398,

%U 10460353202,31381059606,94143178826,282429536478,847288609442,2541865828326,7625597484986,22876792454958,68630377364882,205891132094646,617673396283946,1853020188851838,5559060566555522,16677181699666566

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

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, 1, -3).

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

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

%e 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 + ...

%e 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 + ...

%e such that

%e 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 + ...

%e 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 + ...

%o (PARI) {a(n) = n*polcoeff( log((1 - x)*(1 - x^2)/(1 - 3*x +x*O(x^n))),n)}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A321600, A257970, A105696.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Nov 26 2018