A166343 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+12*x+x^2)/(1-x)^4, read by rows.

1, 1, 1, 1, 12, 1, 1, 27, 27, 1, 1, 58, 162, 58, 1, 1, 121, 718, 718, 121, 1, 1, 248, 2759, 5744, 2759, 248, 1, 1, 503, 9765, 36771, 36771, 9765, 503, 1, 1, 1014, 32816, 205674, 367710, 205674, 32816, 1014, 1, 1, 2037, 106560, 1052408, 3072594, 3072594, 1052408, 106560, 2037, 1

Offset: 1

Author

Roger L. Bagula, Oct 12 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,    1;
  1,   12,      1;
  1,   27,     27,       1;
  1,   58,    162,      58,       1;
  1,  121,    718,     718,     121,       1;
  1,  248,   2759,    5744,    2759,     248,       1;
  1,  503,   9765,   36771,   36771,    9765,     503,      1;
  1, 1014,  32816,  205674,  367710,  205674,   32816,   1014,    1;
  1, 2037, 106560, 1052408, 3072594, 3072594, 1052408, 106560, 2037, 1;
		

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

Crossrefs

Cf. A166340, A166341, A166344, A166345, A166346, A166349.

Cf. A063521, A123125.

Programs

Mathematica

(* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+12*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,4], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)

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 A166343(n,k): return 1 if (k==1) else t(n-1,k,4) - t(n-1,k-1,4)
flatten([[A166343(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

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+12*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)*A063521(k), b(n, 0) = 1, and b(1, k) = 1.

T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 11 2022