A168517 - OEIS

%I #6 Apr 01 2022 09:14:42

%S 1,1,1,1,7,1,1,27,27,1,1,87,260,87,1,1,263,1828,1828,263,1,1,779,

%T 11131,24746,11131,779,1,1,2299,62793,267515,267515,62793,2299,1,1,

%U 6799,338902,2529377,4753074,2529377,338902,6799,1,1,20175,1780242,21935526,70068408,70068408,21935526,1780242,20175,1

%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 = -1, b = 1, and c = 1, read by rows.

%H G. C. Greubel, <a href="/A168517/b168517.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 = -1, b = 1, 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 = -1, b = 1, and c = 1.

%F T(n, n-k) = T(n, k). (End)

%e Triangle begins as:

%e   1;

%e   1,     1;

%e   1,     7,       1;

%e   1,    27,      27,        1;

%e   1,    87,     260,       87,        1;

%e   1,   263,    1828,     1828,      263,        1;

%e   1,   779,   11131,    24746,    11131,      779,        1;

%e   1,  2299,   62793,   267515,   267515,    62793,     2299,       1;

%e   1,  6799,  338902,  2529377,  4753074,  2529377,   338902,    6799,     1;

%e   1, 20175, 1780242, 21935526, 70068408, 70068408, 21935526, 1780242, 20175, 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,-1,1,1], x], {n,0,10}]//Flatten (* modified by _G. C. Greubel_, Mar 31 2022 *)

%o (Sage)

%o def A168517(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([[A168517(n,k,-1,1,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 31 2022

%Y Cf. A168518, 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