A176989 Triangle read by rows: the coefficient [t^n x^k] of n!*(n+2)! *exp(x*t) *(t*(1-2*exp(t))-2*exp(t)) / (2*(1-exp(t))), in row n, k=0..n+1.

2, 2, 5, 6, 3, 2, 40, 24, 8, -1, 30, 300, 120, 30, -12, -24, 360, 2400, 720, 144, 20, -420, -420, 4200, 21000, 5040, 840, 480, 960, -10080, -6720, 50400, 201600, 40320, 5760, -1512, 30240, 30240, -211680, -105840, 635040, 2116800, 362880, 45360, -60480, -120960, 1209600, 806400

Offset

0,1

Comments

Row sums are 4, 14, 74, 479, 3588, 30260, 282720, 2901528, 32598720, 399470400, 5287161600, ...

The expansion of exp(x*t)*(...)/(1-exp(t)) in powers of t starts with a term 2/t, which is ignored and does not enter the table. The coefficient of t^n multiplied by n!(n+2)!/2 defines row n.

Links

Table of n, a(n) for n=0..47.

Formula

exp(x*t)*(t*(1-2*exp(t))-2*exp(t))/(1-exp(t)) = 2/t + (2+2x)*t^0 + (5/3+2x+x^2)*t^1 + ...

Example

The coefficients start in row n=0 with column k=0..n+1 as:
    2,    2;
    5,    6,      3;
    2,   40,     24,     8;
   -1,   30,    300,   120,    30;
  -12,  -24,    360,  2400,   720,    144;
   20, -420,   -420,  4200, 21000,   5040,   840;
  480,  960, -10080, -6720, 50400, 201600, 40320, 5760;

Maple

A176989 := proc(n, k) local x ;
        exp(x*t)*(t*(1-2*exp(t))-2*exp(t))/(1-exp(t))-2/t ;
        n!*(n+2)!/2 *% ; series(%, t, n+3) ;
        convert(%, polynom) ; coeftayl(%, t=0, n) ; coeftayl(%, x=0, k) ;
end proc:
seq (seq(A176989(n, k), k=0..n+1), n=0..5) ; # R. J. Mathar, Dec 20 2010

Mathematica

p[t_] = Exp[x*t]*(t*(1 - 2*Exp[t]) - 2*Exp[t])/(1 - Exp[t]);
a = Table[ CoefficientList[(n!*(n + 2)!/2)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];
Flatten[a]

Crossrefs

Sequence in context: A103892 A000403 A068763 * A367211 A250303 A368554

Adjacent sequences: A176986 A176987 A176988 * A176990 A176991 A176992

Keyword

sign,tabf

Author

Roger L. Bagula, Dec 08 2010

Status

approved