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URL: https://oeis.org/A168551

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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.
5
1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 200, 65, 1, 1, 211, 1536, 1536, 211, 1, 1, 665, 9955, 22350, 9955, 665, 1, 1, 2059, 58521, 251931, 251931, 58521, 2059, 1, 1, 6305, 324322, 2441199, 4596954, 2441199, 324322, 6305, 1, 1, 19171, 1732438, 21480418, 68758180, 68758180, 21480418, 1732438, 19171, 1
OFFSET
0,5
FORMULA
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.
From G. C. Greubel, Mar 31 2022: (Start)
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.
T(n, n-k) = T(n, k). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 5, 1;
1, 19, 19, 1;
1, 65, 200, 65, 1;
1, 211, 1536, 1536, 211, 1;
1, 665, 9955, 22350, 9955, 665, 1;
1, 2059, 58521, 251931, 251931, 58521, 2059, 1;
1, 6305, 324322, 2441199, 4596954, 2441199, 324322, 6305, 1;
1, 19171, 1732438, 21480418, 68758180, 68758180, 21480418, 1732438, 19171, 1;
MATHEMATICA
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]);
Table[CoefficientList[p[x, n, 1, -1, 1], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)
PROG
(SageMath)
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)) )
flatten([[A168552(n, k, 1, -1, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Nov 29 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 31 2022
STATUS
approved