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A377443 Triangular array T(n,k) read by rows, satisfies A377441(n, k+2) = Sum_{m=0..k} T(k, m)*n^m.

%I A377443 #34 Mar 30 2025 20:23:53
%S A377443 2,5,1,14,6,1,42,27,8,1,132,111,45,10,1,429,441,222,67,12,1,1430,1728,
%T A377443 1029,382,93,14,1,4862,6733,4608,2005,599,123,16,1,16796,26181,20199,
%U A377443 10018,3495,881,157,18,1,58786,101763,87270,48445,19188,5641,1236,195,20,1
%N A377443 Triangular array T(n,k) read by rows, satisfies A377441(n, k+2) = Sum_{m=0..k} T(k, m)*n^m.
%F A377443 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 A377443 T(n, 0) = A000108(n+2).
%F A377443 T(n, 1) = A371965(n+2).
%F A377443 T(n, 2) G.f.: x^2*1/( (x - 1)^2*(1 - 4*x)^(3/2) ).
%F A377443 T(n, 3) G.f.: x^3*(3*x - 1)/( (x - 1)^3*(1 - 4*x)^(5/2) ).
%F A377443 T(n, 4) G.f.: x^4*(x^3 + (3*x - 1)^2)/( (x - 1)^4*(1 - 4*x)^(7/2) ).
%F A377443 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 A377443 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 A377443 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 A377443 0 = Sum_{n=0..k} T(n+k, n)*(-1)^n*binomial(k, n).
%F A377443 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 A377443 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 A377443 Triangle T(n, k) starts:
%e A377443 [0]     2
%e A377443 [1]     5,      1
%e A377443 [2]    14,      6,     1
%e A377443 [3]    42,     27,     8,     1
%e A377443 [4]   132,    111,    45,    10,     1
%e A377443 [5]   429,    441,   222,    67,    12,    1
%e A377443 [6]  1430,   1728,  1029,   382,    93,   14,    1
%e A377443 [7]  4862,   6733,  4608,  2005,   599,  123,   16,   1
%e A377443 [8] 16796,  26181, 20199, 10018,  3495,  881,  157,  18,  1
%e A377443 [9] 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1
%o A377443 (PARI)
%o A377443 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 A377443 T(n, k) = Vec(A377441(y, n+5)[n+3])[n-k+1]
%o A377443 (PARI)
%o A377443 A091894(n, k) = 2^(n-2*k-1)*binomial(n-1, 2*k)*(binomial(2*k, k)/(k + 1))
%o A377443 A175136(n, k) = sum(m=0,(n - k)/2,A091894(n-k, m)*binomial(n-m-1, n-k))
%o A377443 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 A377443 Cf. A377441, A377442.
%Y A377443 Cf. A254316 (row sums).
%Y A377443 Cf. A000108, A091894, A175136, A371965.
%K A377443 nonn,tabl
%O A377443 0,1
%A A377443 _Thomas Scheuerle_, Nov 04 2024