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 A350117 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(n+4)) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^(3*n+3)). 1
 1, 1, 5, 43, 443, 5009, 60104, 751778, 9696036, 128037209, 1722632206, 23530913551, 325494250943, 4550333846746, 64189733915195, 912589001283146, 13062908562155459, 188107110626083146, 2723185267618504739, 39610394334267885677 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..19. FORMULA Given g.f. A(x), the following sums are all equal. (1) B(x) = Sum_{n>=0} x^n*A(x)^(1*n) / (1 - x*A(x)^(1*n+4)); (2) B(x) = Sum_{n>=0} x^n*A(x)^(2*n) / (1 - x*A(x)^(3*n+3)); (3) B(x) = Sum_{n>=0} x^n*A(x)^(3*n) / (1 - x*A(x)^(3*n+2)); (4) B(x) = Sum_{n>=0} x^n*A(x)^(4*n) / (1 - x*A(x)^(1*n+1)); (5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+5*n) * (1 - x^2*A(x)^(2*n+5)) / ((1 - x*A(x)^(n+1))*(1 - x*A(x)^(n+4))), (6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(3*n^2+5*n) * (1 - x^2*A(x)^(6*n+5)) / ((1 - x*A(x)^(3*n+2))*(1 - x*A(x)^(3*n+3))); see the example section for the value of B(x). EXAMPLE G.f.: A(x) = 1 + x + 5*x^2 + 43*x^3 + 443*x^4 + 5009*x^5 + 60104*x^6 + 751778*x^7 + 9696036*x^8 + 128037209*x^9 + 1722632206*x^10 + ... such that the following sums are all equal: (1) B(x) = 1/(1 - x*A(x)^4) + x*A(x)^1/(1 - x*A(x)^5) + x^2*A(x)^2/(1 - x*A(x)^6) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^8) + ... (2) B(x) = 1/(1 - x*A(x)^3) + x*A(x)^2/(1 - x*A(x)^6) + x^2*A(x)^4/(1 - x*A(x)^9) + x^3*A(x)^6/(1 - x*A(x)^12) + x^4*A(x)^8/(1 - x*A(x)^15) + ... (3) B(x) = 1/(1 - x*A(x)^2) + x*A(x)^3/(1 - x*A(x)^5) + x^2*A(x)^6/(1 - x*A(x)^8) + x^3*A(x)^9/(1 - x*A(x)^11) + x^4*A(x)^12/(1 - x*A(x)^14) + ... (4) B(x) = 1/(1 - x*A(x)^1) + x*A(x)^4/(1 - x*A(x)^2) + x^2*A(x)^8/(1 - x*A(x)^3) + x^3*A(x)^12/(1 - x*A(x)^4) + x^4*A(x)^16/(1 - x*A(x)^5) + ... where B(x) = 1 + 2*x + 8*x^2 + 51*x^3 + 442*x^4 + 4534*x^5 + 51182*x^6 + 613806*x^7 + 7675397*x^8 + 98971497*x^9 + 1306630823*x^10 + 17575262387*x^11 + 240012293969*x^12 + ... (see A351772). PROG (PARI) {a(n) = my(A=[1, 1, 0]); for(i=0, n, A=concat(A, 0); B1 = sum(m=0, #A, x^m*Ser(A)^(2*m)/(1 - x*Ser(A)^(3*m+3)) ); B2 = sum(m=0, #A, x^m*Ser(A)^(4*m)/(1 - x*Ser(A)^(1*m+1)) ); A[#A-1] = polcoeff((B1 - B2)/2, #A); ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n) = my(A=[1, 1, 0]); for(i=0, n, A=concat(A, 0); B1 = sum(m=0, #A, x^m*Ser(A)^(3*m)/(1 - x*Ser(A)^(3*m+2)) ); B2 = sum(m=0, #A, x^m*Ser(A)^(1*m)/(1 - x*Ser(A)^(1*m+4)) ); A[#A-1] = polcoeff((B1 - B2)/2, #A); ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A340895, A340943, A351772. Sequence in context: A306080 A156886 A112115 * A239265 A369023 A274666 Adjacent sequences: A350114 A350115 A350116 * A350118 A350119 A350120 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 14 2021 STATUS approved

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Last modified August 3 05:17 EDT 2024. Contains 374875 sequences. (Running on oeis4.)