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%I #5 Mar 11 2022 21:24:24
%S 1,1,1,1,2,1,1,7,7,1,1,18,42,18,1,1,41,198,198,41,1,1,88,799,1584,799,
%T 88,1,1,183,2925,10331,10331,2925,183,1,1,374,10056,58874,103310,
%U 58874,10056,374,1,1,757,33160,305888,869794,869794,305888,33160,757,1
%N 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+2*x+x^2)/(1-x)^4, read by rows.
%D Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91
%H G. C. Greubel, <a href="/A166345/b166345.txt">Rows n = 1..50 of the triangle, flattened</a>
%F 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+2*x+x^2)/(1-x)^4.
%F From _G. C. Greubel_, Mar 11 2022: (Start)
%F 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)*A005900(k), b(n, 0) = 1, and b(1, k) = 1.
%F T(n, n-k) = T(n, k). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 7, 7, 1;
%e 1, 18, 42, 18, 1;
%e 1, 41, 198, 198, 41, 1;
%e 1, 88, 799, 1584, 799, 88, 1;
%e 1, 183, 2925, 10331, 10331, 2925, 183, 1;
%e 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1;
%e 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;
%t (* First program *)
%t p[x_, 1]:= x/(1-x)^2;
%t p[x_, 2]:= x*(1+x)/(1-x)^3;
%t p[x_, 3]:= x*(1+10*x+x^2)/(1-x)^4;
%t p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
%t Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
%t (* Second program *)
%t b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
%t 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 T[n_, k_, m_]:= T[n,k,m]= If[k==1, 1, t[n-1,k,m] - t[n-1,k-1,m]];
%t Table[T[n,k,-1], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 11 2022 *)
%o (Sage)
%o def b(n,k,m):
%o if (n<2): return 1
%o elif (k==0): return 0
%o else: return k^(n-1)*((m+3)*k^2 - m)/3
%o @CachedFunction
%o 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 def A166345(n,k): return 1 if (k==1) else t(n-1,k,-1) - t(n-1,k-1,-1)
%o flatten([[A166345(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 11 2022
%Y Cf. A166340, A166341, A166343, A166344, A166346, A166349.
%Y Cf. A005900, A123125.
%K nonn,tabl
%O 1,5
%A _Roger L. Bagula_, Oct 12 2009
%E Edited by _G. C. Greubel_, Mar 11 2022
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