A205495 Convolution related to array A205497 and to generating functions for the rows of the array form of A050446.

1, 46, 937, 12331, 123216, 1019051, 7349140, 47816612, 287357460, 1622135139, 8709442871, 44899559053, 223883501478, 1086005140508, 5148332487873, 23940669359515, 109535136537197, 494307574790201, 2204762394907238, 9736270202183689, 42629974672006973

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 56. Terms corresponding to n=0,...,20 are shown above, with those for n=21,...,60 as follows: {185291835954412064, 800317930217099771, 3438057983187970745, 14700487950597800766, 62602970565114993286, 265668524077091893747, 1124012759249695584332, 4743119424920236606646, 19969635838069446154607, 83911303727287364502524, 351988383031210413076295, 1474320303050934448138586, 6167313972271997160616487, 25770018446823167711177256, 107575128852482376189099657, 448686576996876913475900985, 1870064613139417627428681546, 7789228056784680467763728356, 32425967246106296890368810943, 134922331498272588364476180150, 561170234171421424687450762218, 2333185213162875626980569334586, 9697691681023767935816546925810, 40296761019115897693378020750304, 167405678599573178754554735425500, 695315826495982432201817860350384, 2887471697263577884599209836720724, 11989119731801937435908186367502418, 49773672878387017240820277186133933, 206615368239595050328432096365772786, 857596063782668973911429246019645248, 3559311146445642266628947699835442405, 14771247245703845390492597474797181501, 61297218039066894581942073485999795498, 254355134654745436101804689307395799176, 1055406241452059982356995468881303135245, 4379061349078358899285795579448995148357, 18168834136106060681393826933553149199771, 75380646388163385087709907289615387511431, 312738422596514964765543905180978445030357}.

Links

• L. E. Jeffery, Unit-primitive matrices

Crossrefs

Cf. A205497, A050446, A050447.

Formula

G.f.: F(x) = (1 + 12*x - 112*x^2 - 343*x^3 + 3560*x^4 + 765*x^5 - 40847*x^6 + 10585*x^7 + 310877*x^8 - 193248*x^9 - 1419395*x^10 + 785781*x^11 + 5312667*x^12 - 2323912*x^13 - 15628824*x^14 + 5966469*x^15 + 33782788*x^16 - 10059915*x^17 - 55526776*x^18 + 8186536*x^19 + 73510769*x^20 + 2472617*x^21 - 80001340*x^22 - 15202136*x^23 + 70051834*x^24 + 21752017*x^25 - 47710282*x^26 - 20490103*x^27 + 24620158*x^28 + 14731526*x^29 - 9477868*x^30 - 8317984*x^31 + 2706852*x^32 + 3624852*x^33 - 575397*x^34 - 1176133*x^35 + 88180*x^36 + 269838*x^37 - 5571*x^38 - 39836*x^39 - 2463*x^40 + 2831*x^41 + 1104*x^42 + 107*x^43 - 221*x^44 - 36*x^45 + 23*x^46 + 2*x^47 - x^48) / ((1-x)^6 * (1-x-x^2)^5 * (1-2*x-x^2+x^3)^4 * (1-2*x-3*x^2+x^3+x^4)^3 * (1-3*x-3*x^2+4*x^3+x^4-x^5)^2 * (1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6)).

CONJECTURE 1. a(n) = M_{n,5} = M_{5,n}, where M = A205497.

CONJECTURE 2. Let w=2*cos(Pi/13). Then lim_{n->oo} a(n+1)/a(n) = w^5-4*w^3+3*w = spectral radius of the 6 X 6 unit-primitive matrix (see [Jeffery]) A_{13,5} = [0,0,0,0,0,1; 0,0,0,0,1,1; 0,0,0,1,1,1; 0,0,1,1,1,1; 0,1,1,1,1,1; 1,1,1,1,1,1].