A322191 - OEIS

%I #12 Dec 10 2018 11:54:48

%S 1,3,16,35,141,528,1744,6435,25225,92743,352782,1364216,5200391,

%T 20059504,77744166,300540195,1166803263,4540126119,17672632090,

%U 68923604295,269166933163,1052049488218,4116715364076,16124370433080,63205314207841,247959266485973,973478300504884,3824345303924544,15033633249770955,59132421617169838,232714176627631040,916312070471295267,3609716208975605151,14226520737620321469,56093138920627037034

%N a(n) = coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - (x^(2*n) - y^(2*n))/(x - y)) ), for n >= 1.

%H Paul D. Hanna, <a href="/A322191/b322191.txt">Table of n, a(n) for n = 1..300</a>

%e L.g.f.: L(x) = x + 3*x^2/2 + 16*x^3/3 + 35*x^4/4 + 141*x^5/5 + 528*x^6/6 + 1744*x^7/7 + 6435*x^8/8 + 25225*x^9/9 + 92743*x^10/10 + 352782*x^11/11 + 1364216*x^12/12 + ...

%e RELATED SERIES.

%e Given P(x) = Product_{n>=1} 1/(1 - (x^(2*n) - y^(2*n))/(x - y)),

%e so that P(x) = 1/( (1 - (x^2-y^2)/(x-y)) * (1 - (x^4-y^4)/(x-y)) * (1 - (x^6-y^6)/(x-y)) * (1 - (x^8-y^8)/(x-y)) * (1 - (x^10-y^10)/(x-y)) * ...),

%e then

%e log( P(x) ) = (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2 + (4*x^3 + 6*x^2*y + 6*x*y^2 + 4*y^3)/3 + (1*x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + 1*y^4)/4 + (6*x^5 + 10*x^4*y + 15*x^3*y^2 + 15*x^2*y^3 + 10*x*y^4 + 6*y^5)/5 + (4*x^6 + 12*x^5*y + 24*x^4*y^2 + 32*x^3*y^3 + 24*x^2*y^4 + 12*x*y^5 + 4*y^6)/6 + (8*x^7 + 14*x^6*y + 28*x^5*y^2 + 42*x^4*y^3 + 42*x^3*y^4 + 28*x^2*y^5 + 14*x*y^6 + 8*y^7)/7 + (1*x^8 + 8*x^7*y + 28*x^6*y^2 + 56*x^5*y^3 + 70*x^4*y^4 + 56*x^3*y^5 + 28*x^2*y^6 + 8*x*y^7 + 1*y^8)/8 + (13*x^9 + 27*x^8*y + 63*x^7*y^2 + 123*x^6*y^3 + 171*x^5*y^4 + 171*x^4*y^5 + 123*x^3*y^6 + 63*x^2*y^7 + 27*x*y^8 + 13*y^9)/9 + (6*x^10 + 20*x^9*y + 60*x^8*y^2 + 140*x^7*y^3 + 235*x^6*y^4 + 282*x^5*y^5 + 235*x^4*y^6 + 140*x^3*y^7 + 60*x^2*y^8 + 20*x*y^9 + 6*y^10)/10 + ...

%e in which the coefficients of x^n*y^n/(2*n), for n >= 1, equals

%e [2, 6, 32, 70, 282, 1056, 3488, 12870, 50450, 185486, ...]

%e which is twice this sequence.

%e The exponentiation of the l.g.f. begins

%e exp( L(x) ) = 1 + x + 2*x^2 + 7*x^3 + 16*x^4 + 49*x^5 + 158*x^6 + 480*x^7 + 1565*x^8 + 5372*x^9 + 18168*x^10 + 63018*x^11 + 223069*x^12 + ... + A322192(n)*x^n + ...

%o (PARI) N=35;

%o {L = sum(n=1, N+1, -log(1 - (x^(2*n) - y^(2*n))/(x - y) +O(x^(2*N+1)) +O(y^(2*N+1))) ); }

%o {a(n) = polcoeff( n*polcoeff( L, n, x), n, y)}

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

%Y Cf. A322192.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 10 2018