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A306322 Number of n X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{n,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

%I A306322 #25 Mar 08 2023 03:01:03
%S A306322 1,0,0,25,386,4657,54219,642815,7852836,98755951,1273299491,
%T A306322 16761968919,224508932229,3051075581019,41979207169125,
%U A306322 583745779595077,8192478969914858,115908383594664493,1651636256584103013,23685002515500875105,341589590792856093329
%N A306322 Number of n X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{n,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.
%H A306322 Manuel Kauers and Christoph Koutschan, <a href="/A306322/b306322.txt">Table of n, a(n) for n = 0..836</a> (terms 0..40 from Alois P. Heinz).
%H A306322 M. Kauers and C. 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.
%F A306322 From _Manuel Kauers_ and _Christoph Koutschan_, Mar 02 2023: (Start)
%F A306322 a(n) = 2*Sum_{j=1..n} (Sum_{i=1..n} N'(i, j) + (n-j-1)*N'(j, n)) - 2*binomial(2*n, n) + N'(n, n) + 3 for n>0, where N'(n, k) = (binomial(n+k-1, n-1)*binomial(n+k-1, n))/(n+k-1) denotes the Narayana number N(n+k-1, k).
%F A306322 Recurrence: -2*(n+3)*(n+4)^2*(2*n+7)*(118125*n^10 + 1308375*n^9 + 6016950*n^8 + 14827410*n^7 + 20875365*n^6 + 15986367*n^5 + 4449768*n^4 - 2342808*n^3 - 2279152*n^2 - 660240*n - 64000)*a(n+4) + (n+3)*(10040625*n^13 + 210555000*n^12 + 1942194375*n^11 + 10361592450*n^10 + 35325144315*n^9 + 80085358620*n^8 + 121180651809*n^7 + 117919810482*n^6 + 64349576684*n^5 + 7017979960*n^4 - 14571577344*n^3 - 9566235392*n^2 - 2428639744*n - 221347840)*a(n+3) + (-42170625*n^14 - 981642375*n^13 - 10209053025*n^12 - 62559627795*n^11 - 250621464735*n^10 - 687475711989*n^9 - 1311094658043*n^8 - 1718884004625*n^7 - 1471227292164*n^6 - 691541238960*n^5 - 14462120192*n^4 + 188403075920*n^3 + 108128100864*n^2 + 25779317504*n + 2257059840)*a(n+2) + 2*(2*n+3)*(10040625*n^13 + 215870625*n^12 + 2048259375*n^11 + 11279217825*n^10 + 39828085965*n^9 + 93825035775*n^8 + 147951032109*n^7 + 150478534491*n^6 + 86482913102*n^5 + 11547320420*n^4 - 18788310824*n^3 - 12713618176*n^2 - 3178474112*n - 272670720)*a(n+1) - 8*n*(2*n-1)*(2*n+1)*(2*n+3)*(118125*n^10 + 2489625*n^9 + 23107950*n^8 + 124239510*n^7 + 427851585*n^6 + 984186117*n^5 + 1527319428*n^4 + 1572814284*n^3 + 1022652512*n^2 + 375620224*n + 58236160)*a(n) = 0. (End)
%F A306322 a(n) ~ 25 * 2^(4*n - 3) / (9*Pi*n^2) . - _Vaclav Kotesovec_, Mar 08 2023
%t A306322 Nara[i_, j_] := 1/(i+j-1)*Binomial[i+j-1, i]*Binomial[i+j-1, i-1];
%t A306322 Prepend[Table[2*Sum[Sum[Nara[i, j], {i, n}] + (n-j-1)*Nara[j, n], {j, n}] - 2*Binomial[2*n, n] + Nara[n, n] + 3, {n, 100}], 1] (* _Manuel Kauers_ and _Christoph Koutschan_, Mar 02 2023 *)
%Y A306322 Main diagonal of A323846.
%Y A306322 Column d=2 of A323848.
%K A306322 nonn
%O A306322 0,4
%A A306322 _Alois P. Heinz_, Feb 07 2019