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G.f.: Sum_{n>=0} x^n * Product_{k=3*n..4*n-1} (1 + (1+x)^k).
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%I #5 Oct 20 2020 08:12:18

%S 1,2,7,37,251,1947,17141,167163,1779029,20456722,251997747,3303602265,

%T 45842747207,670308495256,10288071997990,165203260586951,

%U 2767484369773904,48243983405917665,873228719542308898,16378723735806496553,317781161715431909527,6367610715649281689454,131581260677384045968633

%N G.f.: Sum_{n>=0} x^n * Product_{k=3*n..4*n-1} (1 + (1+x)^k).

%F G.f.: Sum_{n>=0} x^n * Product_{k=3*n..4*n-1} (1 + (1+x)^k).

%F G.f.: Sum_{n>=0} x^n * (1+x)^(n*(7*n-1)/2) / ( Product_{k=3*n..4*n} 1 - x*(1+x)^k ).

%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 37*x^3 + 251*x^4 + 1947*x^5 + 17141*x^6 + 167163*x^7 + 1779029*x^8 + 20456722*x^9 + 251997747*x^10 + ...

%e where

%e A(x) = 1 + x*(1 + (1+x)^3) + x^2*(1 + (1+x)^6)*(1 + (1+x)^7) + x^3*(1 + (1+x)^9)*(1 + (1+x)^10)*(1 + (1+x)^11) + x^4*(1 + (1+x)^12)*(1 + (1+x)^13)*(1 + (1+x)^14)*(1 + (1+x)^15) + x^5*(1 + (1+x)^15)*(1 + (1+x)^16)*(1 + (1+x)^17)*(1 + (1+x)^18)*(1 + (1+x)^19) + ... + x^n*Product_{k=3*n..4*n-1} (1 + (1+x)^k) + ...

%e Also

%e A(x) = 1/(1 - x) + x*(1+x)^3/((1 - x*(1+x)^3)*(1 - x*(1+x)^4)) + x^2*(1+x)^13/((1 - x*(1+x)^6)*(1 - x*(1+x)^7)*(1 - x*(1+x)^8)) + x^3*(1+x)^30/((1 - x*(1+x)^9)*(1 - x*(1+x)^10)*(1 - x*(1+x)^11)*(1 - x*(1+x)^12)) + x^4*(1+x)^54/((1 - x*(1+x)^12)*(1 - x*(1+x)^13)*(1 - x*(1+x)^14)*(1 - x*(1+x)^15)*(1 - x*(1+x)^16)) + ... + x^n*(1+x)^(n*(7*n-1)/2)/(Product_{k=3*n..4*n} 1 - x*(1+x)^k) + ...

%o (PARI) {a(n) = polcoeff( sum(m=0, n, x^m * prod(k=3*m,4*m-1,1 + (1+x)^k +x*O(x^n)) ),n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n) = polcoeff( sum(m=0, n, x^m * (1+x +x*O(x^n))^(m*(7*m-1)/2) / prod(k=3*m,4*m, 1 - x*(1+x)^k +x*O(x^n)) ),n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A338179, A338180.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 19 2020