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A214595 T(n,k) = number of n X n X n triangular 0..k arrays with every horizontal row having the same average value.

%I A214595 #18 Aug 21 2025 11:14:51
%S A214595 2,3,2,4,5,2,5,8,23,2,6,13,62,401,2,7,18,157,1862,20351,2,8,25,312,
%T A214595 10177,187862,2869211,2,9,32,601,33352,3330677,63120962,1127599139,2,
%U A214595 10,41,986,103651,20608352,5495329427,71200442882,1248252244661,2,11,50,1619
%N A214595 T(n,k) = number of n X n X n triangular 0..k arrays with every horizontal row having the same average value.
%H A214595 R. H. Hardin, <a href="/A214595/b214595.txt">Table of n, a(n) for n = 1..1475</a>
%F A214595 Empirical for row n:
%F A214595 n=1: a(k)=2*a(k-1)-a(k-2)
%F A214595 n=2: a(k)=2*a(k-1)-2*a(k-3)+a(k-4)
%F A214595 n=3: (order 12 antisymmetric)
%F A214595 n=4: (order 32 symmetric)
%F A214595 n=5: (order 84 symmetric)
%F A214595 T(n, k) = Sum_{s=0..k} Product_{L=2..n} NC(s*L, L, k), where NC(s, n, k) is the number of compositions of sum s with n parts between 0 and k. - _M. F. Hasler_, Aug 21 2025
%e A214595 Table starts
%e A214595 .2.....3......4.......5........6.........7.........8..........9.........10
%e A214595 .2.....5......8......13.......18........25........32.........41.........50
%e A214595 .2....23.....62.....157......312.......601.......986.......1619.......2426
%e A214595 .2...401...1862...10177....33352....103651....250042.....589763....1199614
%e A214595 .2.20351.187862.3330677.20608352.121537201.493575042.1877543213.5767190924
%e A214595 Some solutions for n = k = 4:
%e A214595 .....2........1........2........2........2........2........2........2
%e A214595 ....3.1......0.2......2.2......3.1......2.2......1.3......4.0......4.0
%e A214595 ...3.2.1....0.3.0....3.2.1....2.4.0....0.2.4....3.0.3....1.2.3....4.0.2
%e A214595 ..2.2.3.1..2.1.0.1..1.2.4.1..4.2.2.0..1.4.3.0..4.0.2.2..3.2.3.0..4.0.4.0
%o A214595 (PARI) /* helper function  mult() gives multiplicity of a composition */
%o A214595 mult(p, L=1, m=(#p)!)={for(k=2,#p, p[k]!=p[k-1] && m\=(-L+L=k)!); m\(#p-L+1)!}
%o A214595 A214595(n, k)={sum(a=1,k, prod(L=2,n, my(c=0); forpart(p=L*a, c+=mult(p), [0,k], L); c))+1} \\ _M. F. Hasler_, Aug 21 2025
%Y A214595 Row 2 is A000982(n+1). Other rows: A214596, A214597, A214598.
%Y A214595 Columns: A214589, A214590, A214591, A214592, A214593, A214594.
%K A214595 nonn,tabl,changed
%O A214595 1,1
%A A214595 _R. H. Hardin_, Jul 22 2012