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 A360342 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) for n >= 0. 6
 1, 1, 2, 20, 316, 6686, 173379, 5255624, 180911070, 6938866748, 292678301988, 13446616806957, 668017569348751, 35678261176871802, 2038906890461704040, 124171127134721710130, 8030684434410398312840, 549848454475826567644385, 39744302449387229743134043 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) for n>=0. (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) / (n+1). a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 0.0798993252137..., c = 0.118957192149397... - Vaclav Kotesovec, Feb 06 2023 EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 20*x^3 + 316*x^4 + 6686*x^5 + 173379*x^6 + 5255624*x^7 + 180911070*x^8 + 6938866748*x^9 + ... RELATED SERIES. G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins: B(x) = 1 + x + 3*x^2 + 27*x^3 + 417*x^4 + 8727*x^5 + 225018*x^6 + 6800714*x^7 + 233778499*x^8 + ... + b(n)*x^n + ... such that b(n) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) / (n+1), as well as b(n) = [x^n] A(x)^(n+1) / (n+1), so that b(n) begin: [1/1, 2/2, 9/3, 108/4, 2085/5, 52362/6, 1575126/7, 54405712/8, ...]. ILLUSTRATION OF DEFINITION. The table of coefficients of x^k in A(x)^(n+1) begins: n=0: [1, 1, 2, 20, 316, 6686, 173379, 5255624, ...]; n=1: [1, 2, 5, 44, 676, 14084, 361794, 10897390, ...]; n=2: [1, 3, 9, 73, 1086, 22266, 566441, 16950588, ...]; n=3: [1, 4, 14, 108, 1553, 31312, 788620, 23442284, ...]; n=4: [1, 5, 20, 150, 2085, 41311, 1029745, 30401460, ...]; n=5: [1, 6, 27, 200, 2691, 52362, 1291355, 37859166, ...]; n=6: [1, 7, 35, 259, 3381, 64575, 1575126, 45848685, ...]; n=7: [1, 8, 44, 328, 4166, 78072, 1882884, 54405712, ...]; ... Compare to the table of coefficients in (1 + x*A(x)^(2*n-2))^(n+1): n=0: [1, 1, -2, -1, -32, -519, -11490, -305967, ...]; n=1: [1, 2, 1, 0, 0, 0, 0, 0, ...]; n=2: [1, 3, 9, 28, 180, 2379, 47111, 1182009, ...]; n=3: [1, 4, 22, 108, 745, 8556, 153292, 3658316, ...]; n=4: [1, 5, 40, 265, 2085, 22706, 366450, 8157230, ...]; n=5: [1, 6, 63, 524, 4743, 52362, 781973, 16041192, ...]; n=6: [1, 7, 91, 910, 9415, 109536, 1575126, 29886445, ...]; n=7: [1, 8, 124, 1448, 16950, 211840, 3042820, 54405712, ...]; ... to see that the main diagonals of the tables are the same. PROG (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m-2))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A360343, A360344, A360345, A360346, A360347. Cf. A360231, A302702. Sequence in context: A128481 A367862 A177397 * A323574 A294454 A322729 Adjacent sequences: A360339 A360340 A360341 * A360343 A360344 A360345 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 05 2023 STATUS approved

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Last modified June 21 01:33 EDT 2024. Contains 373535 sequences. (Running on oeis4.)