A377443 Triangular array T(n,k) read by rows, satisfies A377441(n, k+2) = Sum_{m=0..k} T(k, m)*n^m.
2, 5, 1, 14, 6, 1, 42, 27, 8, 1, 132, 111, 45, 10, 1, 429, 441, 222, 67, 12, 1, 1430, 1728, 1029, 382, 93, 14, 1, 4862, 6733, 4608, 2005, 599, 123, 16, 1, 16796, 26181, 20199, 10018, 3495, 881, 157, 18, 1, 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1
Offset: 0
Examples
Triangle T(n, k) starts: [0] 2 [1] 5, 1 [2] 14, 6, 1 [3] 42, 27, 8, 1 [4] 132, 111, 45, 10, 1 [5] 429, 441, 222, 67, 12, 1 [6] 1430, 1728, 1029, 382, 93, 14, 1 [7] 4862, 6733, 4608, 2005, 599, 123, 16, 1 [8] 16796, 26181, 20199, 10018, 3495, 881, 157, 18, 1 [9] 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1
Programs
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PARI
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))) T(n, k) = Vec(A377441(y, n+5)[n+3])[n-k+1]
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PARI
A091894(n, k) = 2^(n-2*k-1)*binomial(n-1, 2*k)*(binomial(2*k, k)/(k + 1)) A175136(n, k) = sum(m=0,(n - k)/2,A091894(n-k, m)*binomial(n-m-1, n-k)) 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)
Formula
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.
T(n, 0) = A000108(n+2).
T(n, 1) = A371965(n+2).
T(n, 2) G.f.: x^2*1/( (x - 1)^2*(1 - 4*x)^(3/2) ).
T(n, 3) G.f.: x^3*(3*x - 1)/( (x - 1)^3*(1 - 4*x)^(5/2) ).
T(n, 4) G.f.: x^4*(x^3 + (3*x - 1)^2)/( (x - 1)^4*(1 - 4*x)^(7/2) ).
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) ).
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) ).
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) ).
0 = Sum_{n=0..k} T(n+k, n)*(-1)^n*binomial(k, n).
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.
T(n+k, n) = Sum_{m=1..k+1} A175136(k+2, k-m+2)*binomial(m+n-1, m-1), for k > 0.