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%I #7 Apr 01 2022 09:15:15
%S 1,1,1,1,3,1,1,11,11,1,1,43,140,43,1,1,159,1244,1244,159,1,1,551,8779,
%T 19954,8779,551,1,1,1819,54249,236347,236347,54249,1819,1,1,5811,
%U 309742,2353021,4440834,2353021,309742,5811,1,1,18167,1684634,21025310,67447952,67447952,21025310,1684634,18167,1
%N 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 = 3, b = -3, and c = 1, read by rows.
%H G. C. Greubel, <a href="/A168552/b168552.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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 = 3, b = -3, and c = 1.
%F From _G. C. Greubel_, Mar 31 2022: (Start)
%F 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 = 3, b = -3, and c = 1.
%F T(n, n-k) = T(n, k). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 11, 11, 1;
%e 1, 43, 140, 43, 1;
%e 1, 159, 1244, 1244, 159, 1;
%e 1, 551, 8779, 19954, 8779, 551, 1;
%e 1, 1819, 54249, 236347, 236347, 54249, 1819, 1;
%e 1, 5811, 309742, 2353021, 4440834, 2353021, 309742, 5811, 1;
%e 1, 18167, 1684634, 21025310, 67447952, 67447952, 21025310, 1684634, 18167, 1;
%t 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 Table[CoefficientList[p[x,n,3,-3,1], x], {n,0,10}]//Flatten (* modified by _G. C. Greubel_, Mar 31 2022 *)
%o (Sage)
%o 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 flatten([[A168552(n,k,3,-3,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 31 2022
%Y Cf. A001263, A168517, A168518, A168549, A168551.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Nov 29 2009
%E Edited by _G. C. Greubel_, Mar 31 2022