A205494 Conjectured row or column n=4 of array A205497.

1, 26, 334, 2951, 20641, 123216, 656683, 3217526, 14786816, 64657546, 271838823, 1107586989, 4399926007, 17122243560, 65514790830, 247212893755, 922136438698, 3406871213836, 12486569116765, 45459575562313, 164578100859837, 593025025473647, 2128399709975819, 7613495897772440

Offset: 0

Author

L. Edson Jeffery, Jan 28 2012

Keywords

• nonn

Comments

The denominator of the generating function for this sequence is a polynomial of degree 35. Terms corresponding to n=0,...,23 are shown above, with those for n=24,...,40 as follows: {27157723973468595, 96643368020414337, 343226612286408932, 1216901732483780905, 4308339945395597755, 15234940157670046379, 53818220864065451564, 189952299613455045068, 669953408386151161398, 2361449534293944339096, 8319329987059336296021, 29296032314800671782284, 103126374236214419873734, 362907786820798388773987, 1276761054260676178577043, 4490840947292979020061377, 15793032895427304036405557}.

See A205497 regarding association of this sequence with generating functions for the rows of the tabular form of A050446.

Links

• L. E. Jeffery, Unit-primitive matrices

Crossrefs

Cf. A205497, A050446, A050447.

Formula

G.f.: (1+4*x-31*x^2 - 67*x^3 + 348*x^4 + 418*x^5 - 1893*x^6 - 1084*x^7 + 4326*x^8 + 4295*x^9 - 7680*x^10 - 9172*x^11 + 9104*x^12 + 11627*x^13 - 5483*x^14 - 10773*x^15 + 1108*x^16 + 7255*x^17 + 315*x^18 - 3085*x^19 - 228*x^20 + 669*x^21 + 102*x^22 - 23*x^23 - 45*x^24 - 16*x^25 + 11*x^26 + 2*x^27 - x^28) / ((1-x)^5 * (1-x-x^2)^4 * (1-2*x-x^2+x^3)^3 * (1-2*x-3*x^2+x^3+x^4)^2 * (1-3*x-3*x^2+4*x^3+x^4-x^5)).

CONJECTURE 1. a(n) = M_{n,4} = M_{4,n}, where M = A205497.

CONJECTURE 2. Let w=2*cos(Pi/11). Then lim_{n->oo} a(n+1)/a(n) = w^4-3*w^2+1 = spectral radius of the 5 X 5 unit-primitive matrix (see [Jeffery]) A_{11,4} = [0,0,0,0,1; 0,0,0,1,1; 0,0,1,1,1; 0,1,1,1,1; 1,1,1,1,1].