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

1,5

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+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)

Example

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;

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 *)

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 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

Crossrefs

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

Cf. A063521, A123125.

Sequence in context: A168646 A051457 A174450 * A186432 A176489 A174039

Adjacent sequences: A166340 A166341 A166342 * A166344 A166345 A166346

Keyword

nonn,tabl

Author

Roger L. Bagula, Oct 12 2009

Extensions

Edited by G. C. Greubel, Mar 11 2022

Status

approved