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A204651 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

%I A204651 #7 Jul 22 2025 18:12:21
%S A204651 8,16,16,28,32,28,48,56,56,48,80,90,104,90,80,132,137,178,178,137,132,
%T A204651 216,200,284,330,284,200,216,352,283,434,571,571,434,283,352,572,390,
%U A204651 637,938,1076,938,637,390,572,928,526,908,1478,1918,1918,1478,908,526,928
%N A204651 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.
%C A204651 Table starts
%C A204651 ...8..16..28...48...80...132...216...352...572....928...1504...2436...3944
%C A204651 ..16..32..56...90..137...200...283...390...526....696....906...1162...1471
%C A204651 ..28..56.104..178..284...434...637...908..1259...1708...2270...2966...3814
%C A204651 ..48..90.178..330..571...938..1478..2248..3317...4766...6690...9198..12415
%C A204651 ..80.137.284..571.1076..1918..3261..5329..8408..12867..19162..27859..39640
%C A204651 .132.200.434..938.1918..3702..6780.11868.19969..32450..51134..78404.117324
%C A204651 .216.283.637.1478.3261..6780.13314.24862.44426..76378.126906.204583.321038
%C A204651 .352.390.908.2248.5329.11868.24862.49312.93219.168960.295101.498776.818748
%H A204651 R. H. Hardin, <a href="/A204651/b204651.txt">Table of n, a(n) for n = 1..6956</a>
%F A204651 Empirical: T(n,k) recurrences
%F A204651 T(1,k)=2*T(1,k-1)-T(1,k-3)
%F A204651 T(2,k)=4*T(2,k-1)-5*T(2,k-2)+5*T(2,k-4)-4*T(2,k-5)+T(2,k-6)
%F A204651 T(3,k)=4*T(3,k-1)-5*T(3,k-2)+5*T(3,k-4)-4*T(3,k-5)+T(3,k-6) for k>7
%F A204651 T(4,k)=5*T(4,k-1)-9*T(4,k-2)+5*T(4,k-3)+5*T(4,k-4)-9*T(4,k-5)+5*T(4,k-6)-T(4,k-7) for k>9
%F A204651 and in general for n>2 (checked to n=15 k=210):
%F A204651 row recurrence coefficients are the coefficients of (1+x)*(1-x)^(k+2) and the row recurrence is valid for k>2*n+1
%e A204651 Some solutions for n=5 k=3
%e A204651 ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..1....0..0..0..1
%e A204651 ..0..0..0..0....0..0..0..1....0..0..0..1....0..0..0..1....0..0..1..1
%e A204651 ..0..0..0..0....0..0..0..1....0..0..0..1....0..0..0..1....0..0..1..1
%e A204651 ..0..0..1..1....0..0..1..1....0..0..0..1....0..0..0..1....0..0..1..1
%e A204651 ..0..0..1..1....1..1..1..1....0..0..0..1....0..0..0..1....0..1..1..1
%e A204651 ..0..0..1..1....1..1..1..1....1..1..1..0....0..1..1..1....0..1..1..1
%K A204651 nonn,tabl
%O A204651 1,1
%A A204651 _R. H. Hardin_ Jan 17 2012