cp's OEIS Frontend

A337841 Triangle read by rows, T(n, k) = binomial(2*n-1, 2*k-1) * binomial(2*n-2*k, n-k) * (k+1) / binomial(n+k+1, n-k) for 0 <= k <= n.

%I A337841 #32 Jun 01 2022 17:41:39
%S A337841 1,0,2,0,3,3,0,6,10,4,0,14,30,21,5,0,36,90,84,36,6,0,99,275,308,180,
%T A337841 55,7,0,286,858,1092,780,330,78,8,0,858,2730,3822,3150,1650,546,105,9,
%U A337841 0,2652,8840,13328,12240,7480,3094,840,136,10,0,8398,29070,46512,46512,31977,15561,5320,1224,171,11
%N A337841 Triangle read by rows, T(n, k) = binomial(2*n-1, 2*k-1) * binomial(2*n-2*k, n-k) * (k+1) / binomial(n+k+1, n-k) for 0 <= k <= n.
%F A337841 T(n,k) = binomial(2*n-1,n-k) * k * (2*k+1) * (2*k+2) / ((n+k)*(n+k+1)) for 1 <= k <= n, and T(n,0) = 0^n for n >= 0.
%F A337841 T(n,n) = n+1 for n >= 0; T(n,n-1) = (n-1) * (2*n-1) for n > 0; T(n,n-2) = (n-1) * (n-2) * (2*n-3) for n > 1.
%F A337841 T(n,k) = T(n-1,k) * (2*n-2) * (2*n-1) / ((n-1) * (n+2) - (k-1) * (k+2)) for 0 <= k < n with initial values T(n,n) = n+1 for n >= 0.
%F A337841 Row sums are A000984(n) for n >= 0.
%F A337841 Alternating row sums are 0 for n > 1.
%F A337841 Sum_{k=0..n} (-1)^k * T(n,k) * (k*(k+1)/2)^m = 0 for 0 <= m <= n-2.
%F A337841 T(n,1) = 12 * binomial(2*n-1,n-1)/((n+1)*(n+2)) = A007054(n) for n > 0.
%F A337841 T(n,k) = T(n,1)*(k*(k+1)*(2*k+1)/6)*binomial(n-1,k-1)/binomial(n+1+k,k-1) for 1 <= k <= n.
%F A337841 From _Werner Schulte_, Nov 09 2020: (Start)
%F A337841 T(n,k) = A128899(n,k) * (k+1) * (2*k+1) / (n+k+1) for 0 <= k <= n.
%F A337841 T(n,0) + Sum_{k=1..n} T(n,k) / (k*(k+1)) = A000108(n) for n >= 0. (End)
%e A337841 The triangle T(n,k) for 0 <= k <= n starts:
%e A337841   n\k:  0     1      2      3      4      5      6     7     8    9  10
%e A337841 =======================================================================
%e A337841    0 :  1
%e A337841    1 :  0     2
%e A337841    2 :  0     3      3
%e A337841    3 :  0     6     10      4
%e A337841    4 :  0    14     30     21      5
%e A337841    5 :  0    36     90     84     36      6
%e A337841    6 :  0    99    275    308    180     55      7
%e A337841    7 :  0   286    858   1092    780    330     78     8
%e A337841    8 :  0   858   2730   3822   3150   1650    546   105     9
%e A337841    9 :  0  2652   8840  13328  12240   7480   3094   840   136   10
%e A337841   10 :  0  8398  29070  46512  46512  31977  15561  5320  1224  171  11
%e A337841 etc.
%p A337841 T := proc(n, k) option remember; if k = n then n+1 else
%p A337841 T(n-1, k)*(2*n-2)*(2*n-1)/((n-1)*(n+2)-(k-1)*(k+2)) fi end:
%p A337841 for n from 0 to 9 do seq(T(n, k), k=0..n) od; # _Peter Luschny_, Nov 02 2020
%Y A337841 Cf. Row sums: A000984, main diagonal: A000027, 1st subdiagonal: A014105, 2nd subdiagonal: A055112, column 0: A000007, column 1: A007054.
%K A337841 nonn,easy,tabl
%O A337841 0,3
%A A337841 _Werner Schulte_, Oct 30 2020