A168523 Triangle of coefficients of g.f. a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = -1, b = 1, c = 1.

1, 1, 1, 1, 8, 1, 1, 31, 31, 1, 1, 98, 290, 98, 1, 1, 289, 1974, 1974, 289, 1, 1, 836, 11719, 25944, 11719, 836, 1, 1, 2419, 64929, 275307, 275307, 64929, 2419, 1, 1, 7046, 346192, 2573466, 4831134, 2573466, 346192, 7046, 1, 1, 20677, 1804144, 22163080, 70723522, 70723522, 22163080, 1804144, 20677, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

From G. C. Greubel, Mar 19 2022: (Start)

G.f.: a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = -1, b = 1, c = 1.

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

Example

Triangle begins as:
  1;
  1,     1;
  1,     8,       1;
  1,    31,      31,        1;
  1,    98,     290,       98,        1;
  1,   289,    1974,     1974,      289,        1;
  1,   836,   11719,    25944,    11719,      836,        1;
  1,  2419,   64929,   275307,   275307,    64929,     2419,       1;
  1,  7046,  346192,  2573466,  4831134,  2573466,   346192,    7046,     1;
  1, 20677, 1804144, 22163080, 70723522, 70723522, 22163080, 1804144, 20677, 1;

Mathematica

T[n_, a_, b_, c_]:= CoefficientList[Series[a*(1+x)^n + b*(1-x)^(n+2)* PolyLog[-n-1, x]/x + 2^n*c*(1-x)^(n+1)*LerchPhi[x, -n, 1/2], {x, 0, 30}], x];
Table[T[n, -1, 1, 1], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 19 2022 *)

Prog

(Sage)
m=12
def LerchPhi(x, s, a): return sum( x^j/(j+a)^s for j in (0..3*m) )
def p(n, x, a, b, c): return a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2)
def T(n, k, a, b, c): return ( p(n, x, a, b, c) ).series(x, n+1).list()[k]
flatten([[T(n, k, -1, 1, 1) for k in (0..n)] for n in (0..m)]) # G. C. Greubel, Mar 19 2022

Crossrefs

Cf. A142458, A142459.

Cf. A168524, A168525.

Sequence in context: A141696 A178122 A142470 * A144439 A157208 A178347

Adjacent sequences: A168520 A168521 A168522 * A168524 A168525 A168526

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 28 2009

Extensions

Edited by G. C. Greubel, Mar 19 2022

Status

approved