login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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

%I #12 Jun 11 2024 19:20:29

%S 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,

%T 10543,0,-1,1,0,-60657,0,259723,0,-2179,0,1,0,-331612,0,4675014,0,

%U -331612,0,1,1,0,-1756340,0,69413294,0,-21707972,0,19673,0,1,0,-9116141,0,906923282,0,-906923282,0,9116141,0,-1

%N 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).

%C What are the imaginary parts? - _N. J. A. Sloane_, Oct 25 2008

%H G. C. Greubel, <a href="/A143196/b143196.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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).

%e Coefficients of p(x, n):

%e 1;

%e 1, i;

%e 1, 6*i, -1;

%e 1, 23*i, -23, -i;

%e 1, 76*i, -230, -76*i, 1;

%e 1, 237*i, -1682, -1682*i, 237, i;

%e 1, 722*i, -10543, -23548*i, 10543, 722*i, -1;

%e 1, 2179*i, -60657, -259723*i, 259723, 60657*i, -2179, -i;

%e Triangle of real coefficients of p(x, n) begins as:

%e 1;

%e 1, 0;

%e 1, 0, -1;

%e 1, 0, -23, 0;

%e 1, 0, -230, 0, 1;

%e 1, 0, -1682, 0, 237, 0;

%e 1, 0, -10543, 0, 10543, 0, -1;

%e 1, 0, -60657, 0, 259723, 0, -2179, 0;

%e 1, 0, -331612, 0, 4675014, 0, -331612, 0, 1;

%e 1, 0, -1756340, 0, 69413294, 0, -21707972, 0, 19673, 0;

%e 1, 0, -9116141, 0, 906923282, 0, -906923282, 0, 9116141, 0, -1;

%t p[x_, n_]:= 2^n*(1-I*x)^(n+1)*LerchPhi[I*x, -n, 1/2];

%t T[n_, k_]:= Coefficient[Series[p[x,n], {x,0,30}], x, k];

%t Table[Re[T[n,k]], {n,0,10}, {k,0,n}]//Flatten

%o (SageMath)

%o def LerchPhi(x, n, q): return sum( x^k/(k+q)^n for k in range(100))

%o def p(n, x): return 2^n*(1-i*x)^(n+1)*LerchPhi(i*x, -n, 1/2)

%o def A143196(n,k): return real( ( p(n, x) ).series(x, n+1).list()[k] )

%o flatten([[A143196(n,k) for k in range(n+1)] for n in (0..12)]) # _G. C. Greubel_, Jun 09 2024

%Y Cf. A060187, A147565.

%K tabl,sign

%O 0,9

%A _Roger L. Bagula_, Oct 19 2008