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 A166344 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+6*x+x^2)/(1-x)^4, read by rows. 5
 1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 34, 90, 34, 1, 1, 73, 406, 406, 73, 1, 1, 152, 1583, 3248, 1583, 152, 1, 1, 311, 5661, 20907, 20907, 5661, 311, 1, 1, 630, 19160, 117594, 209070, 117594, 19160, 630, 1, 1, 1269, 62520, 604496, 1750914, 1750914, 604496, 62520, 1269, 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+6*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)*A000447(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,    6,     1;   1,   15,    15,      1;   1,   34,    90,     34,       1;   1,   73,   406,    406,      73,       1;   1,  152,  1583,   3248,    1583,     152,      1;   1,  311,  5661,  20907,   20907,    5661,    311,     1;   1,  630, 19160, 117594,  209070,  117594,  19160,   630,    1;   1, 1269, 62520, 604496, 1750914, 1750914, 604496, 62520, 1269, 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+6*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 A166344(n, k): return 1 if (k==1) else t(n-1, k, 1) - t(n-1, k-1, 1) flatten([[A166344(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022 CROSSREFS Cf. A166340, A166341, A166343, A166345, A166346, A166349. Cf. A000447, A123125. Sequence in context: A086645 A168291 A154980 * A146766 A176152 A146958 Adjacent sequences:  A166341 A166342 A166343 * A166345 A166346 A166347 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 May 22 19:57 EDT 2022. Contains 353957 sequences. (Running on oeis4.)