cp's OEIS Frontend

A360346 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.

%I A360346 #8 Feb 06 2023 04:19:12
%S A360346 1,1,6,82,1724,47223,1555047,58892186,2496826094,116434989450,
%T A360346 5900151126856,322048641354617,18810964989814291,1169843128503194025,
%U A360346 77145176721564799777,5376524285402806746719,394887654026596322701724,30489608056346314234108286,2469347798211941105406473481
%N A360346 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.
%H A360346 Paul D. Hanna, <a href="/A360346/b360346.txt">Table of n, a(n) for n = 0..300</a>
%F A360346 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A360346 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+2))^(n+1) for n>=0.
%F A360346 (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).
%F A360346 a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 2.153902930660..., c = 0.01676305987174... - _Vaclav Kotesovec_, Feb 06 2023
%e A360346 G.f.: A(x) = 1 + x + 6*x^2 + 82*x^3 + 1724*x^4 + 47223*x^5 + 1555047*x^6 + 58892186*x^7 + 2496826094*x^8 + 116434989450*x^9 + ...
%e A360346 RELATED SERIES.
%e A360346 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A360346 B(x) = 1 + x + 7*x^2 + 101*x^3 + 2161*x^4 + 59544*x^5 + 1965132*x^6 + 74504861*x^7 + 3161424763*x^8 + ... + b(n)*x^n + ...
%e A360346 such that b(n) = [x^n] (1 + x*A(x)^(2*n+2))^(n+1) / (n+1),
%e A360346 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A360346 so that b(n) begin:
%e A360346 [1/1, 2/2, 21/3, 404/4, 10805/5, 357264/6, 13755924/7, 596038888/8, ...].
%e A360346 ILLUSTRATION OF DEFINITION.
%e A360346 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A360346 n=0: [1, 1,  6,   82,  1724,  47223,  1555047,  58892186, ...];
%e A360346 n=1: [1, 2, 13,  176,  3648,  98878,  3231952, 121743878, ...];
%e A360346 n=2: [1, 3, 21,  283,  5790, 155319,  5039055, 188787837, ...];
%e A360346 n=3: [1, 4, 30,  404,  8169, 216924,  6985240, 260270488, ...];
%e A360346 n=4: [1, 5, 40,  540, 10805, 284096,  9079965, 336452690, ...];
%e A360346 n=5: [1, 6, 51,  692, 13719, 357264, 11333293, 417610542, ...];
%e A360346 n=6: [1, 7, 63,  861, 16933, 436884, 13755924, 504036226, ...];
%e A360346 n=7: [1, 8, 76, 1048, 20470, 523440, 16359228, 596038888, ...]; ...
%e A360346 Compare to the table of coefficients in (1 + x*A(x)^(2*n+2))^(n+1):
%e A360346 n=0: [1, 1,   2,   13,   176,   3648, 98878, 3231952, ...];
%e A360346 n=1: [1, 2,   9,   68,   884,  17386,   454318,  14493920, ...];
%e A360346 n=2: [1, 3,  21,  190,  2508,  47406,  1190949,  36928479, ...];
%e A360346 n=3: [1, 4,  38,  404,  5585, 103464,  2504568,  75227160, ...];
%e A360346 n=4: [1, 5,  60,  735, 10805, 200001,  4698210, 136509465, ...];
%e A360346 n=5: [1, 6,  87, 1208, 19011, 357264,  8227591, 231595008, ...];
%e A360346 n=6: [1, 7, 119, 1848, 31199, 602427, 13755924, 376756199, ...];
%e A360346 n=7: [1, 8, 156, 2680, 48518, 970712, 22218108, 596038888, ...]; ...
%e A360346 to see that the main diagonals of the tables are the same.
%o A360346 (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]}
%o A360346 for(n=0, 20, print1(a(n), ", "))
%Y A360346 Cf. A360342, A360343, A360344, A360345, A360347.
%Y A360346 Cf. A302703, A360234.
%K A360346 nonn
%O A360346 0,3
%A A360346 _Paul D. Hanna_, Feb 05 2023