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%I #37 Mar 10 2023 02:25:23
%S 1,2,48,2288,135040,8956752,640160976,48203722464,3772321496064,
%T 304100156874800,25098440923318048,2111488538062121088,
%U 180477438192133215952,15633823902235680250592,1369837117884520736235840,121216041295339359662340288,10819157637786569144853012480
%N Number of permutations of n copies of 1..5 with all adjacent differences <= 1 in absolute value.
%H Alois P. Heinz, <a href="/A177317/b177317.txt">Table of n, a(n) for n = 0..28</a> (terms n=1..16 from R. H. Hardin)
%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. 9-11.
%F From _Manuel Kauers_ and _Christoph Koutschan_, Mar 01 2023: (Start)
%F a(n) = coefficient of x1^n*x2^n*x3^n*x4^n*t^(5*n-1) in (2*t^3*x3(x1*x2+x1*x4*x2+x4*x2+x1*x4)-t^2*x3*(x2*x1-3*x1+x2*x4+x4)-2*t*(x3*x1+x4*x1+x1+x2+x3+x2*x4)+x1+x2+x3+x4+1)/(-t^4*x3*(x1*x2+x1*x4*x2+x4*x2+x1*x4)+t^3*x3*(x2*x1-x1+x2*x4+x4)+t^2*(x3*x1+x4*x1+x1+x2+x3+x2*x4)-t*(x1+x2+x3+x4+1)+1).
%F 3*n^3*(1 + n)*(1 + 3*n)*(2 + 3*n)*(3281160 + 13324928*n + 23607946*n^2 + 23825758*n^3 + 14975281*n^4 + 6000286*n^5 + 1496236*n^6 + 212252*n^7 + 13113*n^8)*a(n) - (1 + n)^2*(14722560 + 163505952*n + 822949992*n^2 + 2464399296*n^3 + 4847819730*n^4 + 6543447222*n^5 + 6186525969*n^6 + 4125650658*n^7 + 1929434771*n^8 + 618883678*n^9 + 129652375*n^10 + 15978026*n^11 + 878571*n^12)*a(n+1) + 2*(2 + n)^2*(20370096 + 207973548*n + 951883014*n^2 + 2588508450*n^3 + 4659341433*n^4 + 5838584798*n^5 + 5211702571*n^6 + 3333874350*n^7 + 1515722000*n^8 + 477646252*n^9 + 99089547*n^10 + 12162378*n^11 + 668763*n^12)*a(n+2) - (2 + n)^2*(3 + n)^4*(10512 + 90060*n + 332910*n^2 + 697266*n^3 + 906481*n^4 + 745834*n^5 + 377636*n^6 + 107348*n^7 + 13113*n^8)*a(n+3) = 0. (End)
%Y Row n=5 of A331562.
%K nonn
%O 0,2
%A _R. H. Hardin_, May 06 2010
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 20 2020
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