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A166341 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.

%I A166341 #7 Mar 11 2022 20:45:19
%S A166341 1,1,1,1,10,1,1,23,23,1,1,50,138,50,1,1,105,614,614,105,1,1,216,2367,
%T A166341 4912,2367,216,1,1,439,8397,31483,31483,8397,439,1,1,886,28264,176314,
%U A166341 314830,176314,28264,886,1,1,1781,91880,903104,2632034,2632034,903104,91880,1781,1
%N A166341 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.
%D A166341 Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91
%H A166341 G. C. Greubel, <a href="/A166341/b166341.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A166341 T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4.
%F A166341 From _G. C. Greubel_, Mar 11 2022: (Start)
%F A166341 T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A007588(k), b(n, 0) = 1, and b(1, k) = 1.
%F A166341 T(n, n-k) = T(n, k). (End)
%e A166341 Triangle begins as:
%e A166341   1;
%e A166341   1,    1;
%e A166341   1,   10,     1;
%e A166341   1,   23,    23,      1;
%e A166341   1,   50,   138,     50,       1;
%e A166341   1,  105,   614,    614,     105,       1;
%e A166341   1,  216,  2367,   4912,    2367,     216,      1;
%e A166341   1,  439,  8397,  31483,   31483,    8397,    439,     1;
%e A166341   1,  886, 28264, 176314,  314830,  176314,  28264,   886,    1;
%e A166341   1, 1781, 91880, 903104, 2632034, 2632034, 903104, 91880, 1781, 1;
%t A166341 (* First program *)
%t A166341 p[x_, 1]:= x/(1-x)^2;
%t A166341 p[x_, 2]:= x*(1+x)/(1-x)^3;
%t A166341 p[x_, 3]:= x*(1+10*x+x^2)/(1-x)^4;
%t A166341 p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
%t A166341 Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
%t A166341 (* Second program *)
%t A166341 b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
%t A166341 t[n_, k_, m_]:= t[n,k,m]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
%t A166341 T[n_, k_, m_]:= T[n,k,m]= If[k==1, 1, t[n-1,k,m] - t[n-1,k-1,m]];
%t A166341 Table[T[n,k,3], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 11 2022 *)
%o A166341 (Sage)
%o A166341 def b(n,k,m):
%o A166341     if (n<2): return 1
%o A166341     elif (k==0): return 0
%o A166341     else: return k^(n-1)*((m+3)*k^2 - m)/3
%o A166341 @CachedFunction
%o A166341 def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
%o A166341 def A166341(n,k): return 1 if (k==1) else t(n-1,k,3) - t(n-1,k-1,3)
%o A166341 flatten([[A166341(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 11 2022
%Y A166341 Cf. A166340, A166343, A166344, A166345, A166346, A166349.
%Y A166341 Cf. A007588, A123125.
%K A166341 nonn,tabl
%O A166341 1,5
%A A166341 _Roger L. Bagula_, Oct 12 2009
%E A166341 Edited by _G. C. Greubel_, Mar 11 2022