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 A317803 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(4*n) )^n = 1. 6
 1, 4, 22, 308, 7877, 287224, 13293116, 735955720, 47105160785, 3410314286768, 275071315285416, 24442342714268592, 2371821148074889444, 249559207019813962752, 28303003280888905543584, 3442273720243525242224992, 446977352681757476329452018, 61724119095080041604018873868, 9033234491867095630258647812994, 1396682556807057529868101744945708, 227509260041431637641628131782970335 (list; graph; refs; listen; history; text; internal format)
 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: A265908 A119009 A326883 * A053722 A336212 A276122 Adjacent sequences: A317800 A317801 A317802 * A317804 A317805 A317806 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 12 2018 STATUS approved

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Last modified November 28 22:51 EST 2022. Contains 358421 sequences. (Running on oeis4.)