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.

1, 0, 0, 25, 386, 4657, 54219, 642815, 7852836, 98755951, 1273299491, 16761968919, 224508932229, 3051075581019, 41979207169125, 583745779595077, 8192478969914858, 115908383594664493, 1651636256584103013, 23685002515500875105, 341589590792856093329

Offset: 0

Author

Alois P. Heinz, Feb 07 2019

Keywords

• nonn

Links

• Manuel Kauers and Christoph Koutschan, Table of n, a(n) for n = 0..836 (terms 0..40 from Alois P. Heinz).
• M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Crossrefs

Main diagonal of A323846.

Column d=2 of A323848.

Programs

Mathematica

Nara[i_, j_] := 1/(i+j-1)*Binomial[i+j-1, i]*Binomial[i+j-1, i-1];
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 *)

Formula

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

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).

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)

a(n) ~ 25 * 2^(4*n - 3) / (9*Pi*n^2) . - Vaclav Kotesovec, Mar 08 2023