A165891 - OEIS

%I #10 Mar 10 2022 02:03:02

%S 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,

%T 1,62,597,2392,5162,6612,5162,2392,597,62,1,1,126,1926,11382,35967,

%U 69132,85492,69132,35967,11382,1926,126,1,1,254,6043,50892,223785,600546,1060411,1277096,1060411,600546,223785,50892,6043,254,1

%N 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.

%H G. C. Greubel, <a href="/A165891/b165891.txt">Rows n = 0..50 of the irregular triangle, flattened</a>

%F 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 ).

%F 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).

%F T(n, n-k) = T(n, k). - _G. C. Greubel_, Mar 09 2022

%e Irregular triangle begins as:

%e   1;

%e   1,   2,    1;

%e   1,   6,   10,     6,     1;

%e   1,  14,   47,    68,    47,    14,     1;

%e   1,  30,  176,   450,   606,   450,   176,    30,     1;

%e   1,  62,  597,  2392,  5162,  6612,  5162,  2392,   597,    62,    1;

%e   1, 126, 1926, 11382, 35967, 69132, 85492, 69132, 35967, 11382, 1926, 126, 1;

%t p[n_, x_]:= p[n, x]= (1/x)*(1+x)^n*(1-x)^(n+2)*PolyLog[-n-1, x];

%t Table[CoefficientList[p[n, x], x], {n,0,12}]//Flatten (* modified by _G. C. Greubel_, Mar 09 2022 *)

%o (Sage)

%o def p(n,x): return (1/x)*(1+x)^n*(1-x)^(n+2)*polylog(-n-1, x)

%o def T(n,k): return ( p(n,x) ).series(x, 2*n+1).list()[k]

%o flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2022

%Y Cf. A158782, A165883, A165889, A165890.

%K nonn,tabf

%O 0,3

%A _Roger L. Bagula_, Sep 29 2009

%E Edited by _G. C. Greubel_, Mar 09 2022