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 A352237 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 3*x*A'(x)). 3
 1, 1, 4, 37, 532, 9994, 226252, 5910445, 173581060, 5634589906, 199792389160, 7671942375898, 316936631324368, 14011781050744984, 660054967923455212, 33008607551445324157, 1746771084107236755604, 97536010045722766992778, 5731874036042145864368824 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA G.f. A(x) satisfies: (1) [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1) for n >= 1. (2) A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)). (3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (3*x). (4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(3*x) dx ). EXAMPLE G.f.: A(x) = 1 + x + 4*x^2 + 37*x^3 + 532*x^4 + 9994*x^5 + 226252*x^6 + 5910445*x^7 + 173581060*x^8 + ... such that A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)). Related table. The table of coefficients of x^k in A(x)^(3*n+1) begins: n=0: [1, 1, 4, 37, 532, 9994, 226252, ...]; n=1: [1, 4, 22, 200, 2717, 48788, 1069122, ...]; n=2: [1, 7, 49, 462, 6069, 104664, 2219784, ...]; n=3: [1, 10, 85, 850, 11020, 183832, 3777355, ...]; n=4: [1, 13, 130, 1391, 18083, 294203, 5869734, ...]; n=5: [1, 16, 184, 2112, 27852, 445632, 8659920, ...]; n=6: [1, 19, 247, 3040, 41002, 650161, 12353059, ...]; ... in which the following pattern holds: [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1), n >= 1, as illustrated by [x^1] A(x)^4 = 4 = [x^0] 4*A(x)^4 = 4*1; [x^2] A(x)^7 = 49 = [x^1] 7*A(x)^7 = 7*7; [x^3] A(x)^10 = 850 = [x^2] 10*A(x)^10 = 10*85; [x^4] A(x)^13 = 18083 = [x^3] 13*A(x)^13 = 13*1391; [x^5] A(x)^16 = 445632 = [x^4] 16*A(x)^16 = 16*27852; [x^6] A(x)^19 = 12353059 = [x^5] 19*A(x)^19 = 19*650161; ... Also, compare the above terms along the diagonal to the series B(x) = A(x*B(x)^3) = 1 + x + 7*x^2 + 85*x^3 + 1391*x^4 + 27852*x^5 + 650161*x^6 + 17204220*x^7 + ... where B(x)^3 = (1/x) * Series_Reversion( x/A(x)^3 ). PROG (PARI) /* Using A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)) */ {a(n) = my(A=1); for(i=1, n, A = 1 + x*A^2/(A - 3*x*A' + x*O(x^n)) ); polcoeff(A, n)} for(n=0, 20, print1(a(n), ", ")) (PARI) /* Using [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1)*A(x)^(3*n+1) */ {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff((x*Ser(A)^(3*(#A)-2) - Ser(A)^(3*(#A)-2)/(3*(#A)-2)), #A-1)); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A088715, A286797, A317352, A352235, A352236, A352238. Sequence in context: A316877 A349254 A277638 * A349714 A121080 A001518 Adjacent sequences: A352234 A352235 A352236 * A352238 A352239 A352240 KEYWORD nonn AUTHOR Paul D. Hanna, Mar 08 2022 STATUS approved

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Last modified March 22 08:21 EDT 2023. Contains 361419 sequences. (Running on oeis4.)