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%I #18 Oct 20 2020 03:37:12
%S 1,1,3,38,931,32226,1415534,74544428,4548135075,314358016202,
%T 24229723093690,2058448459349268,191018823828664206,
%U 19222178481531794836,2084990652057915182076,242526313941563762891992,30119174062262621400424995,3978016516035722365905177306,556837163639960493090777944066,82353722310201117982432743521924
%N G.f. A(x) satisfies: [x^n] A(x)^(2*n+1) / (x*A(x)^2)' = 0 for n>1.
%C Odd terms appear to occur only at positions 0 and 2^n for n>=0.
%C More generally, [x^n] G(x,k)^(k*(n+1)-1) / (x*G(x,k)^k)' = 0 is satisfied by an integer series G(x,k) when k is a fixed positive integer.
%H Paul D. Hanna, <a href="/A300627/b300627.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) satisfies: [x^n] A(x)^(2*n) / (A(x) + 2*x*A'(x)) = 0 for n>1.
%F a(n) ~ c * 2^(3*n) * n! / n^(5/8), where c = 0.0324759424864349046269... - _Vaclav Kotesovec_, Oct 20 2020
%e G.f.: A(x) = 1 + x + 3*x^2 + 38*x^3 + 931*x^4 + 32226*x^5 + 1415534*x^6 + 74544428*x^7 + 4548135075*x^8 + 314358016202*x^9 + ...
%e such that [x^n] A(x)^(2*n+1) / (x*A(x)^2)' = 0 for n>1.
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients in A(x)^(2*n+1) / (x*A(x)^2)' begins:
%e n=0: [1, -3, -6, -203, -6882, -304062, -16218796, -1004084811, ...];
%e n=1: [1, -1, -5, -154, -5629, -259038, -14200498, -896533268, ...];
%e n=2: [1, 1, 0, -89, -4107, -207189, -11944332, -778560170, ...];
%e n=3: [1, 3, 9, 0, -2256, -147537, -9425665, -649322628, ...];
%e n=4: [1, 5, 22, 121, 0, -78928, -6616832, -507901145, ...];
%e n=5: [1, 7, 39, 282, 2753, 0, -3486672, -353291008, ...];
%e n=6: [1, 9, 60, 491, 6111, 90849, 0, -184392528, ...];
%e n=7: [1, 11, 85, 756, 10198, 195493, 3882985, 0, ...]; ...
%e in which the main diagonal consists of all zeros after the initial terms, illustrating that [x^n] A(x)^(2*n+1) / (x*A(x)^2)' = 0 for n>1.
%e RELATED SERIES.
%e A(x)^2 = 1 + 2*x + 7*x^2 + 82*x^3 + 1947*x^4 + 66542*x^5 + 2902550*x^6 + 152184036*x^7 + 9257168147*x^8 + ...
%e (x*A(x)^2)' = 1 + 4*x + 21*x^2 + 328*x^3 + 9735*x^4 + 399252*x^5 + 20317850*x^6 + 1217472288*x^7 + 83314513323*x^8 + ...
%e log(A(x)) = x + 5*x^2/2 + 106*x^3/3 + 3565*x^4/4 + 156126*x^5/5 + 8285474*x^6/6 + 511246324*x^7/7 + 35754687997*x^8/8 + 2786287728022*x^9/9 + ...
%o (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec( Ser(A)^(2*#A-1)/(x*Ser(A)^2)' )[#A]); A[n+1]}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A300994, A300995, A302059, A302060.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 30 2018
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