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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, 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 (list; graph; refs; listen; history; text; internal format)
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 * A250303 A301477 A261895

Adjacent sequences:  A176986 A176987 A176988 * A176990 A176991 A176992

KEYWORD

sign,tabf

AUTHOR

Roger L. Bagula, Dec 08 2010

STATUS

approved

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Last modified May 14 13:50 EDT 2021. Contains 343884 sequences. (Running on oeis4.)