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 A107594 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2-n). 2
 1, 1, 1, 3, 10, 42, 194, 979, 5274, 30037, 179527, 1120612, 7280750, 49120810, 343547469, 2487670468, 18631824735, 144215785791, 1152745117570, 9508011730755, 80861962283808, 708502494881786, 6390084112199801, 59272034375915217, 564899767969587670 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..24. FORMULA G.f. A(x) = x/series-reversion(x*G107595(x)) and thus A(x) = G107595(x/A(x)) where G107595(x) is the g.f. of A107595. G.f. A(x)^2 = x/series-reversion(x*G107596(x)^2) and thus A(x) = G107596(x/A(x)^2) where G107596(x) is the g.f. of A107596. From Paul D. Hanna, Apr 25 2010: (Start) Let A = g.f. A(x), then A satisfies the continued fraction: A = 1/(1- x/(1- (A^2-1)*x/(1- A^4*x/(1- (A^6-A^2)*x/(1- A^8*x/(1- (A^10-A^4)*x/(1- A^12*x/(1- (A^14-A^6)*x/(1- ...))))))))) due to an identity of a partial elliptic theta function. (End) EXAMPLE G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 42*x^5 + 194*x^6 + 979*x^7 +... Let A = A(x) then A = 1 + x*A^0 + x^2*A^2 + x^3*A^6 + x^4*A^12 + x^5*A^20 + x^6*A^30 +... = 1 + x + (x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 27*x^6 + 110*x^7 +...) + (x^3 + 6*x^4 + 21*x^5 + 68*x^6 + 240*x^7 + 948*x^8 + 4140*x^9 +...) + (x^4 + 12*x^5 + 78*x^6 + 388*x^7 + 1737*x^8 + 7632*x^9 +...) + (x^5 + 20*x^6 + 210*x^7 + 1580*x^8 + 9795*x^9 +...) + (x^6 + 30*x^7 + 465*x^8 + 5020*x^9 +...) +... MATHEMATICA m = 25; A[_] = 0; Do[A[x_] = 1 + x + Sum[x^k A[x]^(k^2 - k) + O[x]^j, {k, 2, j}], {j, m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 05 2019 *) PROG (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(k=1, n, A=1+sum(j=1, n, x^j*A^(j^2-j)+x*O(x^n))); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A107590, A107595, A107596. Sequence in context: A094558 A074511 A000249 * A094195 A091843 A361955 Adjacent sequences: A107591 A107592 A107593 * A107595 A107596 A107597 KEYWORD eigen,nonn AUTHOR Paul D. Hanna, May 17 2005 STATUS approved

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Last modified February 24 21:21 EST 2024. Contains 370307 sequences. (Running on oeis4.)