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A143196
Triangle read by rows: real part of Lerch Phi expansion of p(x,n) = 2^n*(1 - i*x)^(n+1) * LerchPhi(i*x, -n, 1/2).
2
1, 1, 0, 1, 0, -1, 1, 0, -23, 0, 1, 0, -230, 0, 1, 1, 0, -1682, 0, 237, 0, 1, 0, -10543, 0, 10543, 0, -1, 1, 0, -60657, 0, 259723, 0, -2179, 0, 1, 0, -331612, 0, 4675014, 0, -331612, 0, 1, 1, 0, -1756340, 0, 69413294, 0, -21707972, 0, 19673, 0, 1, 0, -9116141, 0, 906923282, 0, -906923282, 0, 9116141, 0, -1
OFFSET
0,9
COMMENTS
What are the imaginary parts? - N. J. A. Sloane, Oct 25 2008
FORMULA
T(n, k) = Real Coefficients of p(x, n), where p(x, n) = 2^n*(1-i*x)^(n+1) * LerchPhi(i*x, -n, 1/2).
EXAMPLE
Coefficients of p(x, n):
1;
1, i;
1, 6*i, -1;
1, 23*i, -23, -i;
1, 76*i, -230, -76*i, 1;
1, 237*i, -1682, -1682*i, 237, i;
1, 722*i, -10543, -23548*i, 10543, 722*i, -1;
1, 2179*i, -60657, -259723*i, 259723, 60657*i, -2179, -i;
Triangle of real coefficients of p(x, n) begins as:
1;
1, 0;
1, 0, -1;
1, 0, -23, 0;
1, 0, -230, 0, 1;
1, 0, -1682, 0, 237, 0;
1, 0, -10543, 0, 10543, 0, -1;
1, 0, -60657, 0, 259723, 0, -2179, 0;
1, 0, -331612, 0, 4675014, 0, -331612, 0, 1;
1, 0, -1756340, 0, 69413294, 0, -21707972, 0, 19673, 0;
1, 0, -9116141, 0, 906923282, 0, -906923282, 0, 9116141, 0, -1;
MATHEMATICA
p[x_, n_]:= 2^n*(1-I*x)^(n+1)*LerchPhi[I*x, -n, 1/2];
T[n_, k_]:= Coefficient[Series[p[x, n], {x, 0, 30}], x, k];
Table[Re[T[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten
PROG
(SageMath)
def LerchPhi(x, n, q): return sum( x^k/(k+q)^n for k in range(100))
def p(n, x): return 2^n*(1-i*x)^(n+1)*LerchPhi(i*x, -n, 1/2)
def A143196(n, k): return real( ( p(n, x) ).series(x, n+1).list()[k] )
flatten([[A143196(n, k) for k in range(n+1)] for n in (0..12)]) # G. C. Greubel, Jun 09 2024
CROSSREFS
Sequence in context: A123665 A180729 A119566 * A092993 A238331 A114784
KEYWORD
tabl,sign
AUTHOR
Roger L. Bagula, Oct 19 2008
STATUS
approved