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A141581
Irregular triangle read by rows: T(n, k) = coefficients of f(n, x), where f(n, x) = (1-x)^(2*n+2) * Sum_{k >=0} (k^n * x^k).
1
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, 0, 1, 112, 259, -3808, 8561, -112, -26229, 42432, -26229, -112, 8561, -3808, 259, 112, 1
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.
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
Cf. A123125.
Sequence in context: A061896 A366793 A069850 * A179286 A193690 A108964
KEYWORD
tabf,sign
AUTHOR
EXTENSIONS
Edited and name changed by G. C. Greubel, Sep 16 2024
STATUS
approved