A166340 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+8*x+x^2)/(1-x)^4, read by rows.

1, 1, 1, 1, 8, 1, 1, 19, 19, 1, 1, 42, 114, 42, 1, 1, 89, 510, 510, 89, 1, 1, 184, 1975, 4080, 1975, 184, 1, 1, 375, 7029, 26195, 26195, 7029, 375, 1, 1, 758, 23712, 146954, 261950, 146954, 23712, 758, 1, 1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 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+8*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)*A004466(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,    8,     1;
  1,   19,    19,      1;
  1,   42,   114,     42,       1;
  1,   89,   510,    510,      89,       1;
  1,  184,  1975,   4080,    1975,     184,      1;
  1,  375,  7029,  26195,   26195,    7029,    375,     1;
  1,  758, 23712, 146954,  261950,  146954,  23712,   758,    1;
  1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 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+8*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]= 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, 2], {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 A166340(n, k): return 1 if (k==1) else t(n-1, k, 2) - t(n-1, k-1, 2)
flatten([[A166340(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

Crossrefs

Cf. A166341, A166343, A166344, A166345, A166346, A166349.

Cf. A004466, A123125.

Sequence in context: A051469 A155494 A154227 * A157178 A174303 A176488

Adjacent sequences: A166337 A166338 A166339 * A166341 A166342 A166343

Keyword

nonn,tabl

Author

Roger L. Bagula, Oct 12 2009

Extensions

Edited by G. C. Greubel, Mar 11 2022

Status

approved