The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #31 Nov 17 2024 07:37:00
%S 2,5,1,14,6,1,42,27,8,1,132,111,45,10,1,429,441,222,67,12,1,1430,1728,
%T 1029,382,93,14,1,4862,6733,4608,2005,599,123,16,1,16796,26181,20199,
%U 10018,3495,881,157,18,1,58786,101763,87270,48445,19188,5641,1236,195,20,1
%N Triangular array T(n,k) read by rows, satisfies A377441(n, k+2) = Sum_{m=0..k} T(k, m)*n^m.
%F G.f.: (-(y*x^3-(y+1)*x^2+2*x+1) + sqrt((y*x^3-(y+1)*x^2+x)^2 - 4*(x^3-x^2)*((y+1)*x^2-x)))/(2*(x^3-x^2))/x^2.
%F T(n, 0) = A000108(n+2).
%F T(n, 1) = A371965(n+2).
%F T(n, 2) G.f.: x^2*1/( (x - 1)^2*(1 - 4*x)^(3/2) ).
%F T(n, 3) G.f.: x^3*(3*x - 1)/( (x - 1)^3*(1 - 4*x)^(5/2) ).
%F T(n, 4) G.f.: x^4*(x^3 + (3*x - 1)^2)/( (x - 1)^4*(1 - 4*x)^(7/2) ).
%F T(n, 5) G.f.: x^5*(3*x^3*(3*y - 1) + (3*x - 1)^3)/( (x - 1)^5*(1 - 4*x)^(9/2) ).
%F T(n, 6) G.f.: x^6*(2*x^6 + 6*x^3*(3*x - 1)^2 + (3*x - 1)^4)/( (x - 1)^6*(1 - 4*x)^(11/2) ) ).
%F T(n, 7) G.f.: x^7*(10*x^6*(3*x - 1) + 10*x^3*(3*x - 1)^3 + (3*x - 1)^5)/( (x - 1)^7*(1 - 4*x)^(13/2) ).
%F 0 = Sum_{n=0..k} T(n+k, n)*(-1)^n*binomial(k, n).
%F The diagonal k terms below main diagonal has G.f.: 1 + Sum_{m=1..k+1} A175136(k+2, k-m+2)*(1 - x)^k.
%F T(n+k, n) = Sum_{m=1..k+1} A175136(k+2, k-m+2)*binomial(m+n-1, m-1), for k > 0.
%e Triangle T(n, k) starts:
%e [0] 2
%e [1] 5, 1
%e [2] 14, 6, 1
%e [3] 42, 27, 8, 1
%e [4] 132, 111, 45, 10, 1
%e [5] 429, 441, 222, 67, 12, 1
%e [6] 1430, 1728, 1029, 382, 93, 14, 1
%e [7] 4862, 6733, 4608, 2005, 599, 123, 16, 1
%e [8] 16796, 26181, 20199, 10018, 3495, 881, 157, 18, 1
%e [9] 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1
%o (PARI)
%o A377441(n, max_k) = Vec(-2*((n+1)*x-1)/((x-1)*(n*x-1)+((n*x^2-(n+1)*x+1)^2-4*x*(x-1)*((n+1)*x-1)+O(x^max_k))^(1/2)))
%o T(n, k) = Vec(A377441(y, n+5)[n+3])[n-k+1]
%o (PARI)
%o A091894(n, k) = 2^(n-2*k-1)*binomial(n-1, 2*k)*(binomial(2*k, k)/(k + 1))
%o A175136(n, k) = sum(m=0,(n - k)/2,A091894(n-k, m)*binomial(n-m-1, n-k))
%o T(n, k) = sum(m=1, n+1-k, A175136(n+2-k, n-m+2-k)*binomial(m+k-1, m-1))+(k==0)
%Y Cf. A377441, A377442.
%Y Cf. A254316 (row sums).
%Y Cf. A000108, A091894, A175136, A371965.
%K nonn,tabl,new
%O 0,1
%A _Thomas Scheuerle_, Nov 04 2024