A338185 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) ).

1, 1, 3, 10, 39, 162, 710, 3218, 14975, 71104, 343127, 1677901, 8296177, 41405278, 208316214, 1055403113, 5379799482, 27571260823, 141981480530, 734300071652, 3812386869656, 19862912138493, 103818326606370, 544215209806150, 2860409010422489, 15071419871620972, 79591577077046075

Offset

0,3

Comments

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

(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.

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

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

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 +- ...

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f. A(x) satisfies:

(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) ).

(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) ).

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

a(n) ~ c * d^n / n^(3/2), where d = 5.5920680588399427430141372875417200505037253441108278... and c = 0.40187004264459287738059401335210330541727954857416... - Vaclav Kotesovec, Oct 17 2020

Example

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 + ...
where, if we set A = A(x), then
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)) +- ...
also (trivially),
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)) +- ...
Further,
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) + ...

Prog

(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]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* 1 = A(x) - Sum_{n>=1} x^n * A(x)^(n+1) * Product_{k=1..n-1} (1 - x^k) */
{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)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A338184.

Sequence in context: A307490 A253194 A367258 * A151072 A151073 A363555

Adjacent sequences: A338182 A338183 A338184 * A338186 A338187 A338188

Keyword

nonn

Author

Paul D. Hanna, Oct 16 2020

Status

approved