Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Apr 01 2022 09:14:53
%S 1,1,1,1,12,1,1,51,51,1,1,170,514,170,1,1,521,3646,3646,521,1,1,1552,
%T 22247,49472,22247,1552,1,1,4591,125565,534995,534995,125565,4591,1,1,
%U 13590,677776,5058698,9506078,5058698,677776,13590,1,1,40341,3560448,43870968,140136690,140136690,43870968,3560448,40341,1
%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 = -4, b = 2, and c = 2, read by rows.
%H G. C. Greubel, <a href="/A168518/b168518.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 = -4, b = 2, and c = 2.
%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 = -4, b = 2, and c = 2.
%F T(n, n-k) = T(n, k). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 12, 1;
%e 1, 51, 51, 1;
%e 1, 170, 514, 170, 1;
%e 1, 521, 3646, 3646, 521, 1;
%e 1, 1552, 22247, 49472, 22247, 1552, 1;
%e 1, 4591, 125565, 534995, 534995, 125565, 4591, 1;
%e 1, 13590, 677776, 5058698, 9506078, 5058698, 677776, 13590, 1;
%e 1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 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,-4,2,2], x], {n,0,10}]//Flatten (* modified by _G. C. Greubel_, Mar 31 2022 *)
%o (Sage)
%o def A168518(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([[A168518(n,k,-4,2,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 31 2022
%Y Cf. A142460, A155491, A155495, A157273, A166343.
%Y Cf. A168517, A168549, A168551, A168552.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Nov 28 2009
%E Edited by _G. C. Greubel_, Mar 31 2022