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%I #6 Oct 27 2022 10:19:23
%S 1,1,1,1,4,1,1,13,13,1,1,40,118,40,1,1,121,846,846,121,1,1,364,5279,
%T 11784,5279,364,1,1,1093,30339,129879,129879,30339,1093,1,1,3280,
%U 165820,1242672,2337542,1242672,165820,3280,1,1,9841,878188,10854028,34706710,34706710,10854028,878188,9841,1
%N Triangle, T(n, k) = coefficients [x^k]( p(x, n) ), where p(x, n) = (1/2)*( (1+x)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), read by rows.
%H G. C. Greubel, <a href="/A147565/b147565.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = coefficients [x^k]( p(x, n) ), where p(x, n) = (1/2)*( (1+x)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ).
%F T(n, n-k) = T(n, k).
%F T(n, 1) = A003462(n), n >= 1. - _G. C. Greubel_, Oct 26 2022
%e Triangle of coefficients begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 13, 13, 1;
%e 1, 40, 118, 40, 1;
%e 1, 121, 846, 846, 121, 1;
%e 1, 364, 5279, 11784, 5279, 364, 1;
%e 1, 1093, 30339, 129879, 129879, 30339, 1093, 1;
%e 1, 3280, 165820, 1242672, 2337542, 1242672, 165820, 3280, 1;
%t p[n_, x_]:= (1/2)*((1+x)^n +2^n*(1-x)^(n+1)*LerchPhi[x, -n, 1/2]);
%t Table[CoefficientList[p[n,x], x], {n,0,10}]//Flatten
%o (Magma) // As a triangle
%o LerchPhi:= func< x,n,q | (&+[x^k/(k+q)^n: k in [0..100]]) >;
%o p:= func< n,x | ( (x+1)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) )/2 >;
%o R<x>:=PowerSeriesRing(Integers(), 30);
%o [Coefficients(R!( p(n,x) )): n in [0..12]]; // _G. C. Greubel_, Oct 26 2022
%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 (1/2)*((1+x)^n +2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2))
%o flatten([[( p(n,x) ).series(x,n+1).list()[k] for k in range(n+1)] for n in (0..12)]) # _G. C. Greubel_, Oct 26 2022
%Y Cf. A003462, A147566.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Nov 07 2008
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