A263075 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = n^n * (n-1)! for n>=1.

1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, 20617891510063, 4984425649932314, 1467604324373250545, 517561005098562714944, 215501019188749426210440, 104642607303457024105207408, 58625315029802441203026824094, 37541542090285460025870424920666

Offset: 0

Author

Paul D. Hanna, Oct 08 2015

Keywords

• nonn

Comments

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.]

Examples

			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 +...
The coefficients in A(x)^(n^2) begin:
n=1: [1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, ...];
n=2: [1, 4, 14, 152, 5021, 289824, 24532494, 2841972672, 432284291486, ...];
n=3: [1, 9, 54, 507, 13356, 715635, 58722228, 6685822296, 1005887241243, ...];
n=4: [1, 16, 152, 1536, 31500, 1468016, 114260704, 12668897920, ...];
n=5: [1, 25, 350, 4275, 75000, 2840855, 202155100, 21547156900, ...];
n=6: [1, 36, 702, 10776, 184725, 5598720, 344795598, 34598389248, ...];
n=7: [1, 49, 1274, 24647, 456386, 11753973, 592950960, 54103596918, ...];
n=8: [1, 64, 2144, 51712, 1092016, 26366656, 1071635712, 84557168640, ...];
n=9: [1, 81, 3402, 100791, 2482650, 61309629, 2096140032, 135856780686, ...]; ...
where the terms along the main diagonal begin:
[1, 4, 54, 1536, 75000, 5598720, 592950960, 84557168640, ..., n^n*(n-1)!, ...].
Note that odd terms a(n) occur at positions n starting with:
[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, ...],
which seems to equal A118113, the even Fibbinary numbers + 1, with an initial zero included.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

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

Programs

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]}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ exp(1-exp(-1)) * n! * n^(n-1). - Vaclav Kotesovec, Oct 20 2020