This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345013 #70 Jun 25 2023 09:39:26
%S A345013 1,4,3,15,20,6,56,105,60,10,210,504,420,140,15,792,2310,2520,1260,280,
%T A345013 21,3003,10296,13860,9240,3150,504,28,11440,45045,72072,60060,27720,
%U A345013 6930,840,36
%N A345013 Triangle read by rows, related to clusters of type D.
%C A345013 Let C_{n+1} be the cyclic quiver with n+1 vertices. Empirically, the n-th row is related to the green-mutation partial order on clusters for this quiver, restricted to clusters that do not meet the initial seed.
%C A345013 Apparently, value of the associated polynomials at -2 is A089849, up to sign.
%C A345013 By evaluating the associated polynomials at x-1, one apparently gets A062196.
%C A345013 The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*(x+n+2) / (n! * (n+2)!) in the basis made of the binomial(x+i,i). - _F. Chapoton_, Oct 31 2022
%C A345013 Chapoton's observation above is correct: the precise expansion is (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*(x+n+2) / (n! * (n+2)!) = Sum_{k = 0..n} (-1)^k*T(n+1,k)*binomial(x+2*n+2-k, 2*n+2-k), as can be verified using the WZ algorithm. For example, n = 2 gives (x+1)^2*(x+2)^2*(x+3)*(x+4)/(2!*4!) = 15*binomial(x+6,6) - 20*binomial(x+5,5) + 6*binomial(x+4,4). - _Peter Bala_, Jun 24 2023
%F A345013 T(n, k) = (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n, for n >= 1 and 0 <= k < n.
%F A345013 From _Peter Bala_, Jun 24 2023: (Start)
%F A345013 As conjectured above by Chapoton we have
%F A345013 Sum_{k = 0..n-1} T(n,k)*(x - 1)^k = Sum_{k = 0..n-1} A062196(n-1,k)*x^k and
%F A345013 Sum_{k = 0..n-1} T(n,k)*(-2)^k = (-1)^floor(n/2)*A089849(n) for n >= 1 (both easily verified using the WZ algorithm). (End)
%e A345013 Triangle begins:
%e A345013 [1] 1
%e A345013 [2] 4, 3
%e A345013 [3] 15, 20, 6
%e A345013 [4] 56, 105, 60, 10
%e A345013 [5] 210, 504, 420, 140, 15
%e A345013 [6] 792, 2310, 2520, 1260, 280, 21
%e A345013 [7] 3003, 10296, 13860, 9240, 3150, 504, 28
%e A345013 ...
%o A345013 (Sage)
%o A345013 def T_row(n):
%o A345013 return [(n-k)*binomial(n,k)*binomial(2*n-k,n-1)//n for k in range(n)]
%o A345013 for n in range(1, 8): print(T_row(n))
%o A345013 (PARI) row(n) = vector(n, k, k--; (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n); \\ _Michel Marcus_, Sep 30 2021
%Y A345013 Cf. A001791 (T(n,1)), A000217 (T(n,n)), A026002 (row sums), A000012 (alternating row sum), A051924 (number of clusters of type D_n).
%Y A345013 Cf. A089849, A062196, A063007, A253283.
%K A345013 tabl,nonn
%O A345013 1,2
%A A345013 _F. Chapoton_, Sep 30 2021