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%I #6 Apr 20 2021 12:19:07
%S 46656,466560,4665600,46656000,259200000,1440000000,8000000000,
%T 31500000000,124031250000,488373046875,1531537875000,4802902776000,
%U 15061903105536,40165074948096,107106866528256,285618310742016
%N Number of (n+2) X (5+2) 0..1 arrays with nondecreasing sum of every three consecutive values in every row and column.
%C Column 5 of A251192.
%H R. H. Hardin, <a href="/A251190/b251190.txt">Table of n, a(n) for n = 1..145</a>
%F Empirical: a(n) = 2*a(n-1) -a(n-2) +20*a(n-3) -40*a(n-4) +20*a(n-5) -190*a(n-6) +380*a(n-7) -190*a(n-8) +1140*a(n-9) -2280*a(n-10) +1140*a(n-11) -4845*a(n-12) +9690*a(n-13) -4845*a(n-14) +15504*a(n-15) -31008*a(n-16) +15504*a(n-17) -38760*a(n-18) +77520*a(n-19) -38760*a(n-20) +77520*a(n-21) -155040*a(n-22) +77520*a(n-23) -125970*a(n-24) +251940*a(n-25) -125970*a(n-26) +167960*a(n-27) -335920*a(n-28) +167960*a(n-29) -184756*a(n-30) +369512*a(n-31) -184756*a(n-32) +167960*a(n-33) -335920*a(n-34) +167960*a(n-35) -125970*a(n-36) +251940*a(n-37) -125970*a(n-38) +77520*a(n-39) -155040*a(n-40) +77520*a(n-41) -38760*a(n-42) +77520*a(n-43) -38760*a(n-44) +15504*a(n-45) -31008*a(n-46) +15504*a(n-47) -4845*a(n-48) +9690*a(n-49) -4845*a(n-50) +1140*a(n-51) -2280*a(n-52) +1140*a(n-53) -190*a(n-54) +380*a(n-55) -190*a(n-56) +20*a(n-57) -40*a(n-58) +20*a(n-59) -a(n-60) +2*a(n-61) -a(n-62).
%F Empirical for n mod 3 = 0: a(n) = (1/144603922678272)*n^21 + (25/24100653779712)*n^20 + (1183/16067102519808)*n^19 + (8807/2677850419968)*n^18 + (92569/892616806656)*n^17 + (365155/148769467776)*n^16 + (1496249/33059881728)*n^15 + (407759/612220032)*n^14 + (19465669/2448880128)*n^13 + (285580225/3673320192)*n^12 + (1535707099/2448880128)*n^11 + (1707608047/408146688)*n^10 + (392533463/17006112)*n^9 + (74369693/708588)*n^8 + (369121789/944784)*n^7 + (61815875/52488)*n^6 + (687176/243)*n^5 + (427478/81)*n^4 + (132691/18)*n^3 + 7245*n^2 + 4464*n + 1296.
%F Empirical for n mod 3 = 1: a(n) = (1/144603922678272)*n^21 + (25/24100653779712)*n^20 + (889/12050326889856)*n^19 + (239221/72301961339136)*n^18 + (1686701/16067102519808)*n^17 + (10065229/4016775629952)*n^16 + (4397456/94143178827)*n^15 + (2798441845/4016775629952)*n^14 + (135764613539/16067102519808)*n^13 + (6089990996831/72301961339136)*n^12 + (8369770760039/12050326889856)*n^11 + (114542661774923/24100653779712)*n^10 + (3902341436624303/144603922678272)*n^9 + (254596627215455/2008387814976)*n^8 + (30719033848175/62762119218)*n^7 + (144690929066500/94143178827)*n^6 + (121196895610000/31381059609)*n^5 + (237886638800000/31381059609)*n^4 + (3162196064000000/282429536481)*n^3 + (1101717760000000/94143178827)*n^2 + (726387200000000/94143178827)*n + (681472000000000/282429536481).
%F Empirical for n mod 3 = 2: a(n) = (1/144603922678272)*n^21 + (25/24100653779712)*n^20 + (1183/16067102519808)*n^19 + (237803/72301961339136)*n^18 + (833321/8033551259904)*n^17 + (3288409/1338925209984)*n^16 + (1092133037/24100653779712)*n^15 + (2681321347/4016775629952)*n^14 + (14242869461/1785233613312)*n^13 + (5653090729399/72301961339136)*n^12 + (30480700203433/48201307559424)*n^11 + (34010689997239/8033551259904)*n^10 + (424060067756677/18075490334784)*n^9 + (107723371161847/1004193907488)*n^8 + (134594934647677/334731302496)*n^7 + (1841307277762673/1506290861232)*n^6 + (93062407453967/31381059609)*n^5 + (19537642954025/3486784401)*n^4 + (2243414309843000/282429536481)*n^3 + (747917046310000/94143178827)*n^2 + (156802587200000/31381059609)*n + (419677512800000/282429536481).
%e Some solutions for n=1
%e ..0..0..1..0..1..1..0....0..0..1..0..0..1..0....0..0..0..0..1..1..0
%e ..1..1..1..1..1..1..1....0..1..1..0..1..1..1....1..0..0..1..1..1..1
%e ..0..1..0..0..1..1..1....0..0..0..1..1..0..1....0..0..1..1..0..1..1
%Y Cf. A251192.
%K nonn
%O 1,1
%A _R. H. Hardin_, Nov 30 2014
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