A165889 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( Sum_{j >= 0} j^(n+1)*x^j )^2/x^2, read by rows.

1, 1, 2, 1, 1, 8, 18, 8, 1, 1, 22, 143, 244, 143, 22, 1, 1, 52, 808, 3484, 5710, 3484, 808, 52, 1, 1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1, 1, 240, 16782, 290672, 2000703, 6040992, 8702820, 6040992, 2000703, 290672, 16782, 240, 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)^(2*n+4)*( Sum_{j >= 0} j^(n+1)*x^j )^2/x^2.

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( PolyLog(-n-1, x)/x)^2.

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

Example

Irregular triangle begins as:
  1;
  1,   2,    1;
  1,   8,   18,     8,      1;
  1,  22,  143,   244,    143,     22,      1;
  1,  52,  808,  3484,   5710,   3484,    808,    52,    1;
  1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1;

Mathematica

p[n_, x_]:= p[n, x]= (1/x^2)*(1-x)^(2*n+4)*Sum[k^(n+1)*x^k, {k, 0, Infinity}]^2;
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^2)*(1-x)^(2*n+4)*sum( j^(n+1)*x^j for j in (0..2*n+4) )^2
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)])

Crossrefs

Cf. A165883, A165890, A165891.

Sequence in context: A156901 A167400 A225848 * A087127 A144946 A332717

Adjacent sequences: A165886 A165887 A165888 * A165890 A165891 A165892

Keyword

nonn,tabf

Author

Roger L. Bagula, Sep 29 2009

Extensions

Edited by G. C. Greubel, Mar 09 2022

Status

approved