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URL: https://oeis.org/A166344

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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.
5
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
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
(* Alternative: *)
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
(SageMath)
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Oct 12 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 11 2022
STATUS
approved