A338181 G.f.: A(x) satisfies 1 = Sum_{n>=0} Product_{k=2*n..3*n-1} ( (1+x)^k - A(x) ).

1, 2, 7, 123, 3434, 127389, 5800285, 310671959, 19065484061, 1316721595692, 101007119697478, 8520008641904667, 783858935294161761, 78131288902786790863, 8389163022856205911508, 965540248030539267674460, 118603148642277195505854462, 15489151634920585316431414382, 2143263988200596035777277393467

Offset

0,2

Comments

Compare the g.f. A(x) to functions B(x) and C(x) defined by

A338178: 1 = Sum_{n>=0} Product_{k=n..2*n-1} ((1+x)^k - B(x)) ;

A338183: 1 = Sum_{n>=0} Product_{k=3*n..4*n-1} ((1+x)^k - C(x)).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} Product_{k=2*n..3*n-1} ( (1+x)^k - A(x) ).

(2) 1 = Sum_{n>=0} (1+x)^(n*(5*n-1)/2) / Product_{k=2*n..3*n} (1 + (1+x)^k*A(x)).

a(n) ~ c * d^n * n^n / exp(n), where d = 7.8565736454589... and c = 0.30167199686... - Vaclav Kotesovec, Aug 12 2021

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 123*x^3 + 3434*x^4 + 127389*x^5 + 5800285*x^6 + 310671959*x^7 + 19065484061*x^8 + 1316721595692*x^9 + 101007119697478*x^10 + ...
Let A = A(x), then g.f. A(x) satisfies
1 = 1 + ((1+x)^2 - A) + ((1+x)^4 - A)*((1+x)^5 - A) + ((1+x)^6 - A)*((1+x)^7 - A)*((1+x)^8 - A) + ((1+x)^8 - A)*((1+x)^9 - A)*((1+x)^10 - A)*((1+x)^11 - A) + ((1+x)^10 - A)*((1+x)^11 - A)*((1+x)^12 - A)*((1+x)^13 - A)*((1+x)^14 - A) + ... + Product_{k=2*n..3*n-1} ((1+x)^k - A(x)) + ...
also
1 = 1/(1 + A)  +  (1+x)^2/((1 + (1+x)^2*A)*(1 + (1+x)^3*A))  +  (1+x)^9/((1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A))  +  (1+x)^21/((1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)*(1 + (1+x)^9*A))  +  (1+x)^38/((1 + (1+x)^8*A)*(1 + (1+x)^9*A)*(1 + (1+x)^10*A)*(1 + (1+x)^11*A)*(1 + (1+x)^12*A)) + ... + (1+x)^(n*(5*n-1)/2)/( Product_{k=2*n..3*n} (1 + (1+x)^k*A(x)) ) + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, prod(k=2*m, 3*m-1, (1+x)^k - Ser(A)) ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A338178, A338183.

Sequence in context: A206151 A070521 A292433 * A337764 A000889 A323597

Adjacent sequences: A338178 A338179 A338180 * A338182 A338183 A338184

Keyword

nonn

Author

Paul D. Hanna, Oct 15 2020

Status

approved