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A360237 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+5))^(n+1) for n >= 0.

%I A360237 #23 Feb 05 2023 03:20:14
%S A360237 1,1,7,87,1495,31865,793769,22290228,689397657,23116772771,
%T A360237 831159921411,31787496335409,1285410740283302,54708408148614317,
%U A360237 2441969507507612684,113988651908380638224,5551479742274622439616,281540748098045175486249,14843765603832700589293465
%N A360237 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+5))^(n+1) for n >= 0.
%C A360237 Sequences with g.f. A(x,k) such that [x^n] A(x,k)^(n+1) = [x^n] (1 + x*A(x,k)^(n+k))^(n+1) have a rate of growth: a(n) ~ c(k) * d^n * n! * n^alfa(k), where d = A360279 = 2.1246065836242897918278825746989... (independent on k) and alfa(k) = 1.256334309718765863868089027485828533429844901971596190707510781... + k*0.94499537543174739679595101598799077876961098786349034... - _Vaclav Kotesovec_, Feb 05 2023
%H A360237 Paul D. Hanna, <a href="/A360237/b360237.txt">Table of n, a(n) for n = 0..300</a>
%F A360237 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A360237 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+5))^(n+1) for n>=0.
%F A360237 (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+5))^(n+1) / (n+1).
%F A360237 a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 5.981311186877502847847844107425..., c = 0.000055660090340764345672306890127... - _Vaclav Kotesovec_, Jan 31 2023
%e A360237 G.f.: A(x) = 1 + x + 7*x^2 + 87*x^3 + 1495*x^4 + 31865*x^5 + 793769*x^6 + 22290228*x^7 + 689397657*x^8 + 23116772771*x^9 + ...
%e A360237 RELATED SERIES.
%e A360237 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A360237 B(x) = 1 + x + 8*x^2 + 109*x^3 + 1984*x^4 + 43816*x^5 + 1116182*x^6 + 31810516*x^7 + 994086874*x^8 + ... + b(n)*x^n + ...
%e A360237 such that b(n) = [x^n] (1 + x*A(x)^(n+5))^(n+1) / (n+1),
%e A360237 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A360237 so that b(n) begin:
%e A360237 [1/1, 2/2, 24/3, 436/4, 9920/5, 262896/6, 7813274/7, 254484128/8, ...].
%e A360237 ILLUSTRATION OF DEFINITION.
%e A360237 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A360237 n=0: [1, 1,  7,   87,  1495,  31865,  793769,  22290228, ...];
%e A360237 n=1: [1, 2, 15,  188,  3213,  67938, 1679767,  46874234, ...];
%e A360237 n=2: [1, 3, 24,  304,  5175, 108627, 2666476,  73945899, ...];
%e A360237 n=3: [1, 4, 34,  436,  7403, 154368, 3763020, 103713764, ...];
%e A360237 n=4: [1, 5, 45,  585,  9920, 205626, 4979200, 136401955, ...];
%e A360237 n=5: [1, 6, 57,  752, 12750, 262896, 6325530, 172251150, ...];
%e A360237 n=6: [1, 7, 70,  938, 15918, 326704, 7813274, 211519589, ...];
%e A360237 n=7: [1, 8, 84, 1144, 19450, 397608, 9454484, 254484128, ...]; ...
%e A360237 Compare to the table of coefficients in (1 + x*A(x)^(n+5))^(n+1):
%e A360237 n=0: [1, 1,   5,   45,   585,   9920,   205626,   4979200, ...];
%e A360237 n=1: [1, 2,  13,  126,  1654,  27688,   563565,  13415580, ...];
%e A360237 n=2: [1, 3,  24,  253,  3402,  56679,  1135813,  26574702, ...];
%e A360237 n=3: [1, 4,  38,  436,  6065, 101400,  2008616,  46226504, ...];
%e A360237 n=4: [1, 5,  55,  685,  9920, 167686,  3299580,  74828790, ...];
%e A360237 n=5: [1, 6,  75, 1010, 15285, 262896,  5165838, 115758780, ...];
%e A360237 n=6: [1, 7,  98, 1421, 22519, 396109,  7813274, 173599042, ...];
%e A360237 n=7: [1, 8, 124, 1928, 32022, 578320, 11506804, 254484128, ...]; ...
%e A360237 to see that the main diagonals of the tables are the same.
%o A360237 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+5))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
%o A360237 for(n=0, 20, print1(a(n), ", "))
%Y A360237 Cf. A360231, A302702, A302703, A360234, A360235, A360236.
%K A360237 nonn
%O A360237 0,3
%A A360237 _Paul D. Hanna_, Jan 30 2023