A147565 Triangle, T(n, k) = coefficients [x^k]( p(x, n) ), where p(x, n) = (1/2)*( (1+x)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), read by rows.

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 118, 40, 1, 1, 121, 846, 846, 121, 1, 1, 364, 5279, 11784, 5279, 364, 1, 1, 1093, 30339, 129879, 129879, 30339, 1093, 1, 1, 3280, 165820, 1242672, 2337542, 1242672, 165820, 3280, 1, 1, 9841, 878188, 10854028, 34706710, 34706710, 10854028, 878188, 9841, 1

Offset

0,5

Links

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

Formula

T(n, k) = coefficients [x^k]( p(x, n) ), where p(x, n) = (1/2)*( (1+x)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ).

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

T(n, 1) = A003462(n), n >= 1. - G. C. Greubel, Oct 26 2022

Example

Triangle of coefficients begins as:
  1;
  1,    1;
  1,    4,      1;
  1,   13,     13,       1;
  1,   40,    118,      40,       1;
  1,  121,    846,     846,     121,       1;
  1,  364,   5279,   11784,    5279,     364,      1;
  1, 1093,  30339,  129879,  129879,   30339,   1093,    1;
  1, 3280, 165820, 1242672, 2337542, 1242672, 165820, 3280,   1;

Mathematica

p[n_, x_]:= (1/2)*((1+x)^n +2^n*(1-x)^(n+1)*LerchPhi[x, -n, 1/2]);
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten

Prog

(Magma) // As a triangle
LerchPhi:= func< x, n, q | (&+[x^k/(k+q)^n: k in [0..100]]) >;
p:= func< n, x | ( (x+1)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) )/2 >;
R<x>:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( p(n, x) )): n in [0..12]]; // G. C. Greubel, Oct 26 2022
(SageMath)
def LerchPhi(x, n, q): return sum( x^k/(k+q)^n for k in range(100))
def p(n, x): return (1/2)*((1+x)^n +2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2))
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in range(n+1)] for n in (0..12)]) # G. C. Greubel, Oct 26 2022

Crossrefs

Cf. A003462, A147566.

Sequence in context: A152613 A157153 A212801 * A022167 A339849 A064281

Adjacent sequences: A147562 A147563 A147564 * A147566 A147567 A147568

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 07 2008

Status

approved