A166345 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+2*x+x^2)/(1-x)^4, read by rows.

1, 1, 1, 1, 2, 1, 1, 7, 7, 1, 1, 18, 42, 18, 1, 1, 41, 198, 198, 41, 1, 1, 88, 799, 1584, 799, 88, 1, 1, 183, 2925, 10331, 10331, 2925, 183, 1, 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1, 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 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+2*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)*A005900(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,   2,     1;
  1,   7,     7,      1;
  1,  18,    42,     18,      1;
  1,  41,   198,    198,     41,      1;
  1,  88,   799,   1584,    799,     88,      1;
  1, 183,  2925,  10331,  10331,   2925,    183,     1;
  1, 374, 10056,  58874, 103310,  58874,  10056,   374,   1;
  1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 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+10*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 A166345(n, k): return 1 if (k==1) else t(n-1, k, -1) - t(n-1, k-1, -1)
flatten([[A166345(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

Crossrefs

Cf. A166340, A166341, A166343, A166344, A166346, A166349.

Cf. A005900, A123125.

Sequence in context: A303817 A158200 A220602 * A015110 A128596 A176305

Adjacent sequences: A166342 A166343 A166344 * A166346 A166347 A166348

Keyword

nonn,tabl

Author

Roger L. Bagula, Oct 12 2009

Extensions

Edited by G. C. Greubel, Mar 11 2022

Status

approved