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.

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

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