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A168551 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 1, b = -1, and c = 1, read by rows.

%I A168551 #6 Apr 01 2022 09:15:08
%S A168551 1,1,1,1,5,1,1,19,19,1,1,65,200,65,1,1,211,1536,1536,211,1,1,665,9955,
%T A168551 22350,9955,665,1,1,2059,58521,251931,251931,58521,2059,1,1,6305,
%U A168551 324322,2441199,4596954,2441199,324322,6305,1,1,19171,1732438,21480418,68758180,68758180,21480418,1732438,19171,1
%N A168551 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 1, b = -1, and c = 1, read by rows.
%H A168551 G. C. Greubel, <a href="/A168551/b168551.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168551 G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 1, b = -1, and c = 1.
%F A168551 From _G. C. Greubel_, Mar 31 2022: (Start)
%F A168551 T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = 1, b = -1, and c = 1.
%F A168551 T(n, n-k) = T(n, k). (End)
%e A168551 Triangle begins as:
%e A168551   1;
%e A168551   1,     1;
%e A168551   1,     5,       1;
%e A168551   1,    19,      19,        1;
%e A168551   1,    65,     200,       65,        1;
%e A168551   1,   211,    1536,     1536,      211,        1;
%e A168551   1,   665,    9955,    22350,     9955,      665,        1;
%e A168551   1,  2059,   58521,   251931,   251931,    58521,     2059,       1;
%e A168551   1,  6305,  324322,  2441199,  4596954,  2441199,   324322,    6305,     1;
%e A168551   1, 19171, 1732438, 21480418, 68758180, 68758180, 21480418, 1732438, 19171, 1;
%t A168551 p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
%t A168551 Table[CoefficientList[p[x,n,1,-1,1], x], {n,0,10}]//Flatten (* modified by _G. C. Greubel_, Mar 31 2022 *)
%o A168551 (Sage)
%o A168551 def A168552(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
%o A168551 flatten([[A168552(n,k,1,-1,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 31 2022
%Y A168551 Cf. A001263, A132787.
%Y A168551 Cf. A168517, A168518, A168549, A169552.
%K A168551 nonn,tabl
%O A168551 0,5
%A A168551 _Roger L. Bagula_, Nov 29 2009
%E A168551 Edited by _G. C. Greubel_, Mar 31 2022