A338185 - OEIS

%I #21 Oct 19 2020 06:32:07

%S 1,1,3,10,39,162,710,3218,14975,71104,343127,1677901,8296177,41405278,

%T 208316214,1055403113,5379799482,27571260823,141981480530,

%U 734300071652,3812386869656,19862912138493,103818326606370,544215209806150,2860409010422489,15071419871620972,79591577077046075

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (-1)^n * (n+1) * x^(n*(n+1)/2) * A(x)^(n+1) /( Product_{k=1..n+1} 1 - x^k*A(x) ).

%C Compare the g.f. to the following series identities.

%C (1) 1 = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * a^n /( Product_{k=1..n+1} 1 - x^k*a ), which holds for all a.

%C (2) Q = Sum_{n>=0} (-1)^n * (n+1) * x^(n*(n+1)/2) /( Product_{k=1..n+1} 1 - x^k ),

%C (3) Q = 1 - Sum_{n>=1} x^n * ( Product_{k=1..n-1} 1 - x^k ),

%C where Q = 1 + Sum_{n>=1} (-1)^n * (x^(n*(3*n-1)/2) + x^(n*(3*n+1)/2)) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - x^35 - x^40 +- ...

%H Paul D. Hanna, <a href="/A338185/b338185.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x) satisfies:

%F (1) 1 = Sum_{n>=0} (-1)^n * (n+1) * x^(n*(n+1)/2) * A(x)^(n+1) /( Product_{k=1..n+1} 1 - x^k*A(x) ).

%F (2) 1 = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * A(x)^n /( Product_{k=1..n+1} 1 - x^k*A(x) ).

%F (3) 1 = A(x) - Sum_{n>=1} x^n * A(x)^(n+1) * ( Product_{k=1..n-1} 1 - x^k ).

%F a(n) ~ c * d^n / n^(3/2), where d = 5.5920680588399427430141372875417200505037253441108278... and c = 0.40187004264459287738059401335210330541727954857416... - _Vaclav Kotesovec_, Oct 17 2020

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 162*x^5 + 710*x^6 + 3218*x^7 + 14975*x^8 + 71104*x^9 + 343127*x^10 + 1677901*x^11 + 8296177*x^12 + ...

%e where, if we set A = A(x), then

%e 1 = A/(1 - x*A)  -  2*x*A^2/((1 - x*A)*(1 - x^2*A))  +  3*x^3*A^3/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A))  -  4*x^6*A^4/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A))  +  5*x^10*A^5/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A)*(1 - x^5*A))  -  6*x^15*A^6/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A)*(1 - x^5*A)*(1 - x^6*A)) +- ...

%e also (trivially),

%e 1 = 1/(1 - x*A)  -  x*A/((1 - x*A)*(1 - x^2*A))  +  x^3*A^2/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A))  -  x^6*A^3/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A))  +  x^10*A^4/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A)*(1 - x^5*A))  -  x^15*A^5/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A)*(1 - x^5*A)*(1 - x^6*A)) +- ...

%e Further,

%e 1 = A - x*A^2 - x^2*A^3*(1-x) - x^3*A^4*(1-x)*(1-x^2) - x^4*A^5*(1-x)*(1-x^2)*(1-x^3)*(1-x^4) - x^5*A^6*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5) - x^6*A^7*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6) - ... - x^n * A(x)^(n+1) * Product_{k=0..n-1} (1 - x^k) + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = -Vec( sum(m=0,sqrtint(2*n+1), (-1)^m * (m+1) * x^(m*(m+1)/2) * Ser(A)^(m+1) / prod(k=1,m+1,1 - x^k*Ser(A))) )[#A] );A[n+1]}

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

%o (PARI) /* 1 = A(x) - Sum_{n>=1} x^n * A(x)^(n+1) * Product_{k=1..n-1} (1 - x^k) */

%o {a(n) = my(A=1); for(i=1,n, A = 1 + sum(m=1,n, x^m * A^(m+1) * prod(k=1,m-1, 1 - x^k +x*O(x^n)) )); polcoeff(A,n)}

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

%Y Cf. A338184.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 16 2020