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A263190 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = (n!)^2 for n>=1.

%I A263190 #12 Nov 01 2017 21:59:26
%S A263190 1,1,0,1,46,1723,81104,4793304,349869074,31080492631,3318717525832,
%T A263190 421195540320465,62871475566985208,10927921934497456588,
%U A263190 2191068343727736627744,502384409006686040020572,130687814451798554601790746,38294333521028379285810681487,12557951067433973525611840784048,4581888866092825667058378205370595
%N A263190 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = (n!)^2 for n>=1.
%C A263190 CONJECTURES.
%C A263190 (1) Limit a(n)/(n!)^2 = 1/exp(1).
%C A263190 (2) There are no negative terms.
%C A263190 (3) ODD TERMS: It appears that for k>0, a(k) is odd iff k = 2*A003714(n)+1 for n>=0, where A003714 is the fibbinary numbers (integers whose binary representation contains no consecutive ones); this is true for at least the first 531 terms. [See also A171791 and A263075.]
%C A263190 Conjectures hold to at least a(1000). - _Sean A. Irvine_, Oct 21 2015
%H A263190 Paul D. Hanna, <a href="/A263190/b263190.txt">Table of n, a(n) for n = 0..265</a>
%e A263190 G.f.: A(x) = 1 + x + x^3 + 46*x^4 + 1723*x^5 + 81104*x^6 + 4793304*x^7 +...
%e A263190 The coefficients in A(x)^(n^2) begin:
%e A263190 n=1: [1, 1, 0, 1, 46, 1723, 81104, 4793304, 349869074, ...];
%e A263190 n=2: [1, 4, 6, 8, 197, 7456, 345654, 20167888, 1458010566, ...];
%e A263190 n=3: [1, 9, 36, 93, 612, 19197, 866208, 49440834, 3515499819, ...];
%e A263190 n=4: [1, 16, 120, 576, 2796, 44656, 1803872, 99433344, ...];
%e A263190 n=5: [1, 25, 300, 2325, 14400, 130705, 3606800, 183492150, ...];
%e A263190 n=6: [1, 36, 630, 7176, 61821, 518400, 8260086, 332807184, ...];
%e A263190 n=7: [1, 49, 1176, 18473, 216482, 2154775, 25401600, 655445812, ...];
%e A263190 n=8: [1, 64, 2016, 41728, 642352, 8045248, 95405312, 1625702400, ...];
%e A263190 n=9: [1, 81, 3240, 85401, 1673946, 26315199, 360707040, 5266837404, 131681894400, ...]; ...
%e A263190 where the terms along the main diagonal begin:
%e A263190 [1, 4, 36, 576, 14400, 518400, 25401600, 1625702400, ..., (n!)^2, ...].
%e A263190 LOCATION OF ODD TERMS.
%e A263190 Note that odd terms a(n) occur at positions n starting with:
%e A263190 [0, 1, 3, 5, 9, 11, 17, 19, 21, 33, 35, 37, 41, 43, 65, 67, 69, 73, 75, 81, 83, 85, 129, 131, 133, 137, 139, 145, 147, 149, 161, 163, 165, 169, 171, 257, ...],
%e A263190 which seems to equal A118113, the even fibbinary numbers + 1, with an initial zero included.
%o A263190 (PARI) {a(n) = local(A=[1, 1]); for(i=1, n+1, A=concat(A, 0); m=#A; A[m] = ( m!^2 - Vec(Ser(A)^(m^2))[m] )/m^2 ); A[n+1]}
%o A263190 for(n=0, 20, print1(a(n), ", "))
%Y A263190 Cf. A263075, A294360, A171791, A003714, A118113.
%K A263190 nonn
%O A263190 0,5
%A A263190 _Paul D. Hanna_, Oct 12 2015