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 A166345 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. 5
 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, 88, 1, 1, 183, 2925, 10331, 10331, 2925, 183, 1, 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1, 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 REFERENCES Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91 LINKS G. C. Greubel, Rows n = 1..50 of the triangle, flattened FORMULA 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. From G. C. Greubel, Mar 11 2022: (Start) 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. T(n, n-k) = T(n, k). (End) EXAMPLE Triangle begins as: 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, 88, 1; 1, 183, 2925, 10331, 10331, 2925, 183, 1; 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1; 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1; MATHEMATICA (* First program *) p[x_, 1]:= x/(1-x)^2; p[x_, 2]:= x*(1+x)/(1-x)^3; p[x_, 3]:= x*(1+10*x+x^2)/(1-x)^4; p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x] Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n, 12}]//Flatten (* Second program *) b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]]; 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[n_, k_, m_]:= T[n, k, m]= If[k==1, 1, t[n-1, k, m] - t[n-1, k-1, m]]; Table[T[n, k, -1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Mar 11 2022 *) PROG (Sage) def b(n, k, m): if (n<2): return 1 elif (k==0): return 0 else: return k^(n-1)*((m+3)*k^2 - m)/3 @CachedFunction def t(n, k, m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n, j, m) for j in (0..k) ) def A166345(n, k): return 1 if (k==1) else t(n-1, k, -1) - t(n-1, k-1, -1) flatten([[A166345(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022 CROSSREFS Cf. A166340, A166341, A166343, A166344, A166346, A166349. Cf. A005900, A123125. Sequence in context: A303817 A158200 A220602 * A015110 A128596 A176305 Adjacent sequences: A166342 A166343 A166344 * A166346 A166347 A166348 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Oct 12 2009 EXTENSIONS Edited by G. C. Greubel, Mar 11 2022 STATUS approved

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Last modified December 1 15:29 EST 2022. Contains 358468 sequences. (Running on oeis4.)