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A360236 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+4))^(n+1) for n >= 0.

%I A360236 #19 Feb 02 2023 21:10:38
%S A360236 1,1,6,66,1028,20138,464863,12162876,351915528,11075859686,
%T A360236 374858234365,13530279602015,517628371405448,20890826296067329,
%U A360236 886175281852068632,39393952245422498344,1830781283537184304756,88768944166701791039297,4482797026386165709436753,235417696462456105986818505
%N A360236 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+4))^(n+1) for n >= 0.
%H A360236 Paul D. Hanna, <a href="/A360236/b360236.txt">Table of n, a(n) for n = 0..300</a>
%F A360236 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A360236 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+4))^(n+1) for n>=0.
%F A360236 (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+4))^(n+1) / (n+1).
%F A360236 a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 5.036315811445755451051893091437..., c = 0.000317937301879544729612100255927... - _Vaclav Kotesovec_, Jan 31 2023
%e A360236 G.f.: A(x) = 1 + x + 6*x^2 + 66*x^3 + 1028*x^4 + 20138*x^5 + 464863*x^6 + 12162876*x^7 + 351915528*x^8 + 11075859686*x^9 + ...
%e A360236 RELATED SERIES.
%e A360236 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A360236 B(x) = 1 + x + 7*x^2 + 85*x^3 + 1401*x^4 + 28339*x^5 + 666638*x^6 + 17651052*x^7 + 514911165*x^8 + ... + b(n)*x^n + ...
%e A360236 such that b(n) = [x^n] (1 + x*A(x)^(n+4))^(n+1) / (n+1),
%e A360236 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A360236 so that b(n) begin:
%e A360236 [1/1, 2/2, 21/3, 340/4, 7005/5, 170034/6, 4666466/7, 141208416/8, ...].
%e A360236 ILLUSTRATION OF DEFINITION.
%e A360236 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A360236 n=0: [1, 1,  6,  66,  1028,  20138,  464863,  12162876, ...];
%e A360236 n=1: [1, 2, 13, 144,  2224,  43124,  986694,  25632830, ...];
%e A360236 n=2: [1, 3, 21, 235,  3606,  69264, 1571169,  40527480, ...];
%e A360236 n=3: [1, 4, 30, 340,  5193,  98888, 2224444,  56974172, ...];
%e A360236 n=4: [1, 5, 40, 460,  7005, 132351, 2953185,  75110670, ...];
%e A360236 n=5: [1, 6, 51, 596,  9063, 170034, 3764599,  95085882, ...];
%e A360236 n=6: [1, 7, 63, 749, 11389, 212345, 4666466, 117060623, ...];
%e A360236 n=7: [1, 8, 76, 920, 14006, 259720, 5667172, 141208416, ...]; ...
%e A360236 Compare to the table of coefficients in (1 + x*A(x)^(n+4))^(n+1):
%e A360236 n=0: [1, 1,   4,   30,   340,   5193,   98888,   2224444, ...];
%e A360236 n=1: [1, 2,  11,   90,  1025,  15330,  284912,   6277922, ...];
%e A360236 n=2: [1, 3,  21,  190,  2220,  32862,  597579,  12884601, ...];
%e A360236 n=3: [1, 4,  34,  340,  4131,  61208, 1094268,  23093756, ...];
%e A360236 n=4: [1, 5,  50,  550,  7005, 104951, 1856360,  38416740, ...];
%e A360236 n=5: [1, 6,  69,  830, 11130, 170034, 2996425,  61005672, ...];
%e A360236 n=6: [1, 7,  91, 1190, 16835, 263956, 4666466,  93880165, ...];
%e A360236 n=7: [1, 8, 116, 1640, 24490, 395968, 7067220, 141208416, ...]; ...
%e A360236 to see that the main diagonals of the tables are the same.
%o A360236 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+4))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
%o A360236 for(n=0, 20, print1(a(n), ", "))
%Y A360236 Cf. A360231, A302702, A302703, A360234, A360235, A360237.
%K A360236 nonn
%O A360236 0,3
%A A360236 _Paul D. Hanna_, Jan 30 2023