A205496 Convolution related to array A205497 and to generating functions for the rows of the array form of A050446.
1, 79, 2475, 47191, 656683, 7349140, 70148989, 593513485, 4571277561, 32672880245, 219830952888, 1407595988962, 8650512982826, 51368774778763, 296342413123845, 1668132449230997, 9195464663247238, 49787415018534288, 265430586786327769
Offset: 0
Keywords
Links
- L. E. Jeffery, Unit-primitive matrices
Formula
G.f.: F(x) = (1 + 29*x - 330*x^2 - 1870*x^3 + 28792*x^4 - 28880*x^5 - 658872*x^6 + 1808035*x^7 + 7251417*x^8 - 30049286*x^9 - 53844318*x^10 + 331611771*x^11 + 172019006*x^12 - 2314667923*x^13 - 44340353*x^14 + 12301024850*x^15 - 283356562*x^16 - 53520778564*x^17 + 21918429228*x^18 + 188280737400*x^19 - 99256863420*x^20 - 537933519143*x^21 + 304479953092*x^22 + 1292735746371*x^23 - 685767992532*x^24 - 2703731985407*x^25 + 1220124121648*x^26 + 4969059486596*x^27 - 1817137951816*x^28 - 7940770334300*x^29 + 2310666239334*x^30 + 10897173663437*x^31 - 242841325861*x^32 - 12794627581139*x^33 + 1919519246791*x^34 + 12918502357203*x^35 - 852890650171*x^36 -11317650709986*x^37 - 313858871781*x^38 + 8665013739391*x^39 + 1068808054156*x^40 - 5804674396693*x^41 - 1231795216164*x^42 + 3382179875958*x^43 + 984955686298*x^44 - 1694171598050*x^45 - 619939090864*x^46 + 718589694092*x^47 + 323730198889*x^48 - 253619875999*x^49 - 144187648137*x^50 + 72968474423*x^51 + 55421646471*x^52 - 16658211415*x^53 - 18346712946*x^54 + 2894246774*x^55 + 5160729532*x^56 - 351795527*x^57 - 1206372119*x^58 + 22006791*x^59 + 227332930*x^60 + 1758161*x^61 - 33060926*x^62 - 881244*x^63 + 3436739*x^64 + 218431*x^65 - 208580*x^66 - 43625*x^67 - 299*x^68 + 6491*x^69 + 1284*x^70 - 646*x^71 - 104*x^72 + 38*x^73 +3*x^74 -x^75) / ((1-x)^7 * (1-x-x^2)^6 * (1-2*x-x^2+x^3)^5 * (1-2*x-3*x^2+x^3+x^4)^4 * (1-3*x-3*x^2+4*x^3+x^4-x^5)^3 * (1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6)^2 * (1-4*x-6*x^2+10*x^3+5*x^4-6*x^5-x^6+x^7)).
CONJECTURE 1. a(n) = M_{n,6} = M_{6,n}, where M = A205497.
CONJECTURE 2. Let w=2*cos(Pi/15). Then lim_{n->oo} a(n+1)/a(n) = w^6-5*w^4+6*w^2-1 = spectral radius of the 7 X 7 unit-primitive matrix (see [Jeffery]) A_{15,6} = [0,0,0,0,0,0,1; 0,0,0,0,0,1,1; 0,0,0,0,1,1,1; 0,0,0,1,1,1,1; 0,0,1,1,1,1,1; 0,1,1,1,1,1,1; 1,1,1,1,1,1,1].
Comments