A165891 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.

1, 1, 2, 1, 1, 6, 10, 6, 1, 1, 14, 47, 68, 47, 14, 1, 1, 30, 176, 450, 606, 450, 176, 30, 1, 1, 62, 597, 2392, 5162, 6612, 5162, 2392, 597, 62, 1, 1, 126, 1926, 11382, 35967, 69132, 85492, 69132, 35967, 11382, 1926, 126, 1, 1, 254, 6043, 50892, 223785, 600546, 1060411, 1277096, 1060411, 600546, 223785, 50892, 6043, 254, 1

Offset

0,3

Links

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

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ).

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

T(n, n-k) = T(n, k). - G. C. Greubel, Mar 09 2022

Example

Irregular triangle begins as:
  1;
  1,   2,    1;
  1,   6,   10,     6,     1;
  1,  14,   47,    68,    47,    14,     1;
  1,  30,  176,   450,   606,   450,   176,    30,     1;
  1,  62,  597,  2392,  5162,  6612,  5162,  2392,   597,    62,    1;
  1, 126, 1926, 11382, 35967, 69132, 85492, 69132, 35967, 11382, 1926, 126, 1;

Mathematica

p[n_, x_]:= p[n, x]= (1/x)*(1+x)^n*(1-x)^(n+2)*PolyLog[-n-1, x];
Table[CoefficientList[p[n, x], x], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 09 2022 *)

Prog

(Sage)
def p(n, x): return (1/x)*(1+x)^n*(1-x)^(n+2)*polylog(-n-1, x)
def T(n, k): return ( p(n, x) ).series(x, 2*n+1).list()[k]
flatten([[T(n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2022

Crossrefs

Cf. A158782, A165883, A165889, A165890.

Sequence in context: A144089 A172107 A349226 * A039763 A094262 A123554

Adjacent sequences: A165888 A165889 A165890 * A165892 A165893 A165894

Keyword

nonn,tabf

Author

Roger L. Bagula, Sep 29 2009

Extensions

Edited by G. C. Greubel, Mar 09 2022

Status

approved