|
%I #18 Dec 07 2019 00:51:22
%S 1,1,1,2,2,1,4,4,3,1,7,7,6,4,1,11,11,10,8,5,1,16,16,15,13,10,6,1,22,
%T 22,21,19,16,12,7,1,29,29,28,26,23,19,14,8,1,37,37,36,34,31,27,22,16,
%U 9,1,46,46,45,43,40,36,31,25,18,10,1,56,56,55,53,50,46,41,35,28,20,11,1
%N Triangle read by rows: T(n,k) = ((n - k)*(n + k - 1) + 2)/2, 0 <= k <= n.
%C This triangle equals A309559 with reversed rows and supplemented main diagonal (all terms are 1).
%C There are two lower triangular matrices M and N so that the matrix product M * N equals T (seen as a matrix).
%C / 1 \ / 1 \
%C | 0 1 | | 1 1 |
%C | 0 1 1 | | 1 1 1 |
%C M(n,k) = | 0 1 2 1 | N(n,k) = | 1 1 1 1 |
%C | 0 1 2 3 1 | | 1 1 1 1 1 |
%C | 0 1 2 3 4 1 | | 1 1 1 1 1 1 |
%C \ . . . . . . . / \ . . . . . . . /
%C The matrix product N * M equals the rascal triangle A077028 (seen as a matrix).
%F O.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = ((t^2+(1-t)^2) * (1-x*t) + x * t^2 * (1-t)) / ((1-t)^3 * (1-x*t)^2).
%F G.f. of column k: Sum_{n>=k} T(n,k) * t^n = t^k * (t^2/(1-t)^3 + 1/(1-t) + k*t/(1-t)^2) for k >= 0.
%F T(n,k) = 1 + T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 0 < k < n with initial values T(n,0) = (n*(n-1)+2)/2 and T(n,n) = 1 for n >= 0.
%F T(n,k) = (2 + T(n-1,k-1) * T(n-1,k+1)) / T(n-2,k) for 0 < k < n-1 with initial values given above and T(n,n-1) = n for n > 0.
%F Referring to the triangle M(n,k) (see comments), we get:
%F (1) Sum_{k=0..n} (k+1) * M(n,k) = A116731(n+1) for n >= 0;
%F (2) Sum_{k=1..n} k * M(n,k) = A081489(n) for n >= 1.
%F T(n,k) = T(n-1,k-1) + n-k for 0 < k <= n with initial values T(n,0) = (n*(n-1)+2)/2 for n >= 0.
%F T(n,k) = 2 * T(n-1,k-1) - T(n-2,k-2) for 1 < k <= n with initial values T(0,0) = 1 and T(n,0) = T(n,1) = (n*(n-1)+2)/2 for n > 0.
%e The triangle T(n,k) starts:
%e n \ k : 0 1 2 3 4 5 6 7 8 9 10 11
%e ==================================================================
%e 0 : 1
%e 1 : 1 1
%e 2 : 2 2 1
%e 3 : 4 4 3 1
%e 4 : 7 7 6 4 1
%e 5 : 11 11 10 8 5 1
%e 6 : 16 16 15 13 10 6 1
%e 7 : 22 22 21 19 16 12 7 1
%e 8 : 29 29 28 26 23 19 14 8 1
%e 9 : 37 37 36 34 31 27 22 16 9 1
%e 10 : 46 46 45 43 40 36 31 25 18 10 1
%e 11 : 56 56 55 53 50 46 41 35 28 20 11 1
%e etc.
%Y Row sums equal A116731(n+1).
%Y Row sums apart from column 0 equal A081489.
%Y Cf. A077028, A309559.
%K nonn,tabl
%O 0,4
%A _Werner Schulte_, Nov 22 2019
|
