OFFSET
0,5
COMMENTS
Former title: Cosine projection of Eulerian numbers (A123125) as a coefficient triangle: f(x,n) = cos(w) * (1 - 2*cos(w)*x + x^2)^(n+1) * Sum_{k >= 0} (k^n * x^k), where w = 0.
REFERENCES
Douglas C. Montgomery and Lynwood A, Johnson, Forecasting and Time Series Analysis, McGraw-Hill, New York,1976, page 91.
LINKS
G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
FORMULA
T(n, k) = coefficients of f(n, x, w), where f(n, x, w) = cos(w) * (1 - 2*cos(w)*x + x^2)^(n+1) * Sum_{k >=0} k^n * x^k. This sequence is the case of w = 0.
Sum_{k=0..(2*n+1)} T(n, k) = 0 (row sums).
EXAMPLE
Irregular triangle begins as:
1, -1;
0, 1, -2, 1;
0, 1, -2, 0, 2, -1;
0, 1, 0, -9, 16, -9, 0, 1;
0, 1, 6, -34, 46, 0, -46, 34, -6, -1;
0, 1, 20, -75, 0, 330, -552, 330, 0, -75, 20, 1;
0, 1, 50, -76, -650, 2325, -2652, 0, 2652, -2325, 650, 76, -50, -1;
MATHEMATICA
w = 0;
f[x_, n_]:= f[x, n]= (1 - 2*Cos[w]*x + x^2)^(n+1)*Sum[k^n*x^k*Cos[w], {k, 0, Infinity}];
Table[CoefficientList[f[x, n], x], {n, 0, 10}]//Flatten
PROG
(Magma)
m:=12;
R<x>:=PowerSeriesRing(Rationals(), 2*m+1);
f:= func< n, x | (1-x)^(2*n+2)*(&+[j^n*x^j: j in [0..2*m+4]]) >;
A141581:= func< n, k | Coefficient(R!( f(n, x) ), k) >;
[A141581(n, k): k in [0..2*n+1], n in [0..m-2]]; // G. C. Greubel, Sep 16 2024
(SageMath)
m=12
def f(n, x): return (1-x)^(2*n+2)*sum(j^n*x^j for j in range(2*m+4))
def A141581(n, k):
P.<x> = PowerSeriesRing(ZZ, 2*m+4)
return P( f(n, x) ).list()[k]
flatten([[ A141581(n, k) for k in range(2*n+2)] for n in range(m-2)]) # G. C. Greubel, Sep 16 2024
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Roger L. Bagula and Gary W. Adamson, Aug 19 2008
EXTENSIONS
Edited and name changed by G. C. Greubel, Sep 16 2024
STATUS
approved