A168552 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 = 3, b = -3, and c = 1, read by rows.

1, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 43, 140, 43, 1, 1, 159, 1244, 1244, 159, 1, 1, 551, 8779, 19954, 8779, 551, 1, 1, 1819, 54249, 236347, 236347, 54249, 1819, 1, 1, 5811, 309742, 2353021, 4440834, 2353021, 309742, 5811, 1, 1, 18167, 1684634, 21025310, 67447952, 67447952, 21025310, 1684634, 18167, 1

Offset

0,5

Links

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

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 = 3, b = -3, 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 = 3, b = -3, and c = 1.

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

Example

Triangle begins as:
  1;
  1,     1;
  1,     3,       1;
  1,    11,      11,        1;
  1,    43,     140,       43,        1;
  1,   159,    1244,     1244,      159,        1;
  1,   551,    8779,    19954,     8779,      551,        1;
  1,  1819,   54249,   236347,   236347,    54249,     1819,       1;
  1,  5811,  309742,  2353021,  4440834,  2353021,   309742,    5811,     1;
  1, 18167, 1684634, 21025310, 67447952, 67447952, 21025310, 1684634, 18167, 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, 3, -3, 1], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)

Prog

(Sage)
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, 3, -3, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

Crossrefs

Cf. A001263, A168517, A168518, A168549, A168551.

Sequence in context: A348211 A176468 A176421 * A111473 A234944 A067402

Adjacent sequences: A168549 A168550 A168551 * A168553 A168554 A168555

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 29 2009

Extensions

Edited by G. C. Greubel, Mar 31 2022

Status

approved