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%I #20 Mar 10 2023 02:25:40
%S 0,2,12,90,556,5242,42380,479306,4817484,63779034,767504524,
%T 11661506218,163541678156,2806878055610,44960579074956,
%U 860568917787402,15502269624912460,327460573946510746,6552868832109180044,151436547414562736234,3332986639447590230604,83655126041771262574458
%N Permanent of the (0,1)-matrix with ij-th entry equal to zero iff (i=1,j=1),(i=1,j=2),(i=1,j=3),(i=2,j=3),(i=3,j=3),... In other words, the ij-th entry of the matrix is zero iff it is on the path which start from the entry (i=1,j=1) and moves in the matrix alternating 3 steps to the right to 3 steps down.
%H Vaclav Kotesovec, <a href="/A098926/b098926.txt">Table of n, a(n) for n = 3..36</a>
%H Manuel Kauers and Christoph Koutschan, <a href="https://arxiv.org/abs/2303.02793">Some D-finite and some Possibly D-finite Sequences in the OEIS</a>, arXiv:2303.02793 [cs.SC], 2023, pp. 43-44.
%F From _Manuel Kauers_ and _Christoph Koutschan_, Mar 02 2023: (Start)
%F Conjectured recurrence of order 8 and degree 7: (3*n^5 + 80*n^4 + 763*n^3 + 3184*n^2 + 5915*n + 4080)*a(n+8) + (-3*n^5 - 83*n^4 - 833*n^3 - 3663*n^2 - 6967*n - 4465)*a(n+7) + (-3*n^7 - 122*n^6 - 2039*n^5 - 18038*n^4 - 90333*n^3 - 252920*n^2 - 364438*n - 211080)*a(n+6) + (17*n^5 + 445*n^4 + 4253*n^3 + 17161*n^2 + 24893*n + 1765)*a(n+5) + (9*n^7 + 318*n^6 + 4409*n^5 + 30672*n^4 + 113879*n^3 + 219268*n^2 + 186788*n + 35600)*a(n+4) + (-n^5 + 103*n^4 + 2125*n^3 + 14395*n^2 + 38283*n + 32845)*a(n+3) + (-9*n^7 - 294*n^6 - 3677*n^5 - 22722*n^4 - 76591*n^3 - 146304*n^2 - 157554*n - 81720)*a(n+2) - (n+1)*(13*n^4 + 388*n^3 + 3717*n^2 + 13424*n + 16865)*a(n+1) + n*(n+1)*(3*n^5 + 95*n^4 + 1113*n^3 + 5983*n^2 + 14907*n + 14025)*a(n) = 0.
%F Conjectured differential equation for the generating function: x^5*(x^2 - 1)^3*(x^8 - 2*x^7 - 12*x^6 + 28*x^5 - 10*x^4 - 22*x^3 + 4*x^2 + 4*x + 1)*f'''(x) + x^4*(x^2 - 1)*(5*x^12 - 12*x^11 - 76*x^10 + 212*x^9 - 143*x^8 - 156*x^7 + 80*x^6 + 36*x^5 + 7*x^4 - 24*x^3 - 4*x^2 + 8*x + 3)*f''(x) + x*(x^2 - 1)*(4*x^14 - 9*x^13 - 95*x^12 + 342*x^11 - 288*x^10 - 29*x^9 - 75*x^8 - 336*x^7 + 252*x^6 + 121*x^5 - 53*x^4 + 10*x^3 - 3*x - 1)*f'(x) + 2*(2*x^14 - 13*x^13 + 12*x^12 - 128*x^11 + 40*x^10 + 513*x^9 - 410*x^8 - 24*x^7 + 218*x^6 - 283*x^5 + 16*x^4 + 104*x^3 - 4*x^2 - 9*x - 2)*f(x) = 0. (End)
%F Conjecture: a(n) ~ exp(-2) * n!, based on the recurrence by _Manuel Kauers_ and _Christoph Koutschan_. - _Vaclav Kotesovec_, Mar 02 2023
%e a(3) = 12 because 12 is the permanent of the following 5 X 5 matrix:
%e [0 0 0 1 1]
%e [1 1 0 1 1]
%e [1 1 0 0 0]
%e [1 1 1 1 0]
%e [1 1 1 1 0]
%o (PARI) a(n)={my(M=matrix(n, n, i, j, j-i<>1 && (i%2==0 || abs(j-i-1)<>1))); matpermanent(M)} \\ _Andrew Howroyd_, Nov 05 2019
%K nonn
%O 3,2
%A _Simone Severini_, Oct 19 2004
%E Terms a(11) and beyond from _Andrew Howroyd_, Nov 05 2019
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