A098926 - OEIS

A098926

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.

0, 2, 12, 90, 556, 5242, 42380, 479306, 4817484, 63779034, 767504524, 11661506218, 163541678156, 2806878055610, 44960579074956, 860568917787402, 15502269624912460, 327460573946510746, 6552868832109180044, 151436547414562736234, 3332986639447590230604, 83655126041771262574458

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OFFSET

3,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 3..36

Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 43-44.

FORMULA

From Manuel Kauers and Christoph Koutschan, Mar 02 2023: (Start)

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.

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)

Conjecture: a(n) ~ exp(-2) * n!, based on the recurrence by Manuel Kauers and Christoph Koutschan. - Vaclav Kotesovec, Mar 02 2023

EXAMPLE

a(3) = 12 because 12 is the permanent of the following 5 X 5 matrix:

[0 0 0 1 1]

[1 1 0 1 1]

[1 1 0 0 0]

[1 1 1 1 0]

PROG

(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

CROSSREFS

Sequence in context: A174356 A355138 A121357 * A074610 A372109 A250130

Adjacent sequences: A098923 A098924 A098925 * A098927 A098928 A098929