A263075 - OEIS

%I #25 Oct 20 2020 04:00:41

%S 1,1,2,31,1150,68713,5914776,692005074,105932315154,20617891510063,

%T 4984425649932314,1467604324373250545,517561005098562714944,

%U 215501019188749426210440,104642607303457024105207408,58625315029802441203026824094,37541542090285460025870424920666

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

%C 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 A263190 and A171791.]

%H Paul D. Hanna, <a href="/A263075/b263075.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ exp(1-exp(-1)) * n! * n^(n-1). - _Vaclav Kotesovec_, Oct 20 2020

%e G.f.: A(x) = 1 + x + 2*x^2 + 31*x^3 + 1150*x^4 + 68713*x^5 + 5914776*x^6 + 692005074*x^7 + 105932315154*x^8 +...

%e The coefficients in A(x)^(n^2) begin:

%e n=1: [1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, ...];

%e n=2: [1, 4, 14, 152, 5021, 289824, 24532494, 2841972672, 432284291486, ...];

%e n=3: [1, 9, 54, 507, 13356, 715635, 58722228, 6685822296, 1005887241243, ...];

%e n=4: [1, 16, 152, 1536, 31500, 1468016, 114260704, 12668897920, ...];

%e n=5: [1, 25, 350, 4275, 75000, 2840855, 202155100, 21547156900, ...];

%e n=6: [1, 36, 702, 10776, 184725, 5598720, 344795598, 34598389248, ...];

%e n=7: [1, 49, 1274, 24647, 456386, 11753973, 592950960, 54103596918, ...];

%e n=8: [1, 64, 2144, 51712, 1092016, 26366656, 1071635712, 84557168640, ...];

%e n=9: [1, 81, 3402, 100791, 2482650, 61309629, 2096140032, 135856780686, ...]; ...

%e where the terms along the main diagonal begin:

%e [1, 4, 54, 1536, 75000, 5598720, 592950960, 84557168640, ..., n^n*(n-1)!, ...].

%e Note that odd terms a(n) occur at positions n starting with:

%e [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 which seems to equal A118113, the even Fibbinary numbers + 1, with an initial zero included.

%o (PARI) {a(n) = local(A=[1,1]); for(i=1,n+1, A=concat(A,0); m=#A; A[m] = ( m^m*(m-1)! - Vec(Ser(A)^(m^2))[m] )/m^2 );A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A171791, A263190, A003714, A118113, A244589.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 08 2015