A317803 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n) )^n = 1.

1, 4, 22, 308, 7877, 287224, 13293116, 735955720, 47105160785, 3410314286768, 275071315285416, 24442342714268592, 2371821148074889444, 249559207019813962752, 28303003280888905543584, 3442273720243525242224992, 446977352681757476329452018, 61724119095080041604018873868, 9033234491867095630258647812994, 1396682556807057529868101744945708, 227509260041431637641628131782970335

Offset

0,2

Links

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

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n) )^n.

(2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+4) )^n.

(3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+4) )^n / (1+x)^(4*n+4).

(4) Let B(x,p) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n + p) )^n ,

then B(x,p) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*(n+1)) )^n / (1+x)^((4-p)*(n+1)), where B(x,0) = 1 and B(x,4) = A(x).

a(n) ~ 2^(2*n - log(2)/8 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018

Example

G.f.: A(x) = 1 + 4*x + 22*x^2 + 308*x^3 + 7877*x^4 + 287224*x^5 + 13293116*x^6 + 735955720*x^7 + 47105160785*x^8 + 3410314286768*x^9 + 275071315285416*x^10 + ...
such that
1 = 1  +  (1/A(x) - 1/(1+x)^4)  +  (1/A(x) - 1/(1+x)^8)^2  +  (1/A(x) - 1/(1+x)^12)^3  +  (1/A(x) - 1/(1+x)^16)^4  +  (1/A(x) - 1/(1+x)^20)^5  +  (1/A(x) - 1/(1+x)^24)^6  +  (1/A(x) - 1/(1+x)^28)^7  +  (1/A(x) - 1/(1+x)^32)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - 1/(1+x)^8)  +  (1/A(x) - 1/(1+x)^12)^2  +  (1/A(x) - 1/(1+x)^16)^3  +  (1/A(x) - 1/(1+x)^20)^4  +  (1/A(x) - 1/(1+x)^24)^5  +  (1/A(x) - 1/(1+x)^28)^6  +  (1/A(x) - 1/(1+x)^32)^7  +  (1/A(x) - 1/(1+x)^36)^8  + ...
RELATED SERIES.
(1) The series B(x,1) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+1) )^n begins
B(x,1) = 1 + x + 4*x^2 + 58*x^3 + 1482*x^4 + 53953*x^5 + 2496149*x^6 + 138245508*x^7 + 8853719964*x^8 + 641386920943*x^9 + 51762649442019*x^10 + ...
where B(x,1) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+4) )^n / (1+x)^(3*n+3).
(2) The series B(x,2) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+2) )^n begins
B(x,2) = 1 + 2*x + 9*x^2 + 128*x^3 + 3270*x^4 + 119002*x^5 + 5502295*x^6 + 304531768*x^7 + 19491119849*x^8 + 1411222743454*x^9 + 113839065423087*x^10 + ...
where B(x,2) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+4) )^n / (1+x)^(2*n+2).
(3) The series B(x,3) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+3) )^n begins
B(x,3) = 1 + 3*x + 15*x^2 + 211*x^3 + 5392*x^4 + 196341*x^5 + 9079538*x^6 + 502467023*x^7 + 32153605481*x^8 + 2327561975059*x^9 + 187722580703289*x^10 + ...
where B(x,3) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n+4) )^n / (1+x)^(n+1).

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - 1/(1+x +x*O(x^#A))^(4*m+4) )^m ) )[#A]/2 ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A317339, A317801, A317802, A317995, A317668.

Sequence in context: A359111 A119009 A326883 * A053722 A336212 A276122

Adjacent sequences: A317800 A317801 A317802 * A317804 A317805 A317806

Keyword

nonn

Author

Paul D. Hanna, Aug 12 2018

Status

approved