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Table T(n,m) read by rows: the coefficient of [t^n x^m] of 2*n!*(n+2)!*exp(x*t)*( t*(1-exp(t))-exp(t) ) / (1-exp(t) ), 0<=m<=n+1.
0

%I #7 Mar 12 2014 16:37:17

%S 2,4,13,6,6,0,104,24,16,-2,0,780,120,60,0,-48,0,6240,720,288,40,0,

%T -840,0,54600,5040,1680,0,1920,0,-13440,0,524160,40320,11520,-3024,0,

%U 60480,0,-211680,0,5503680,362880,90720,0,-241920,0,1612800,0,-3386880,0,62899200,3628800,806400,604800,0,-11975040,0,39916800,0,-55883520,0,778377600,39916800,7983360,0,72576000,0,-479001600,0,958003200,0,-958003200,0,10378368000,479001600,87091200

%N Table T(n,m) read by rows: the coefficient of [t^n x^m] of 2*n!*(n+2)!*exp(x*t)*( t*(1-exp(t))-exp(t) ) / (1-exp(t) ), 0<=m<=n+1.

%C Row sums are: 6, 25, 144, 958, 7200, 60520, 564480, 5803056, 65318400, 798940800,

%C 10538035200,....

%C The bivariate expansion is exp(x*t)*( t*(1-exp(t))-exp(t) )/ (1-exp(t) )= 1/t +(1/2+x)*t^0 + (13/12+x/2+x^2/2)*t +... The leading term 1/t corresponding to n = -1 is dropped, and the other rows are scaled with a factor 2*n!*(n+2)!.

%e 2, 4;

%e 13, 6, 6;

%e 0, 104, 24, 16;

%e -2, 0, 780, 120, 60;

%e 0, -48, 0, 6240, 720, 288;

%e 40, 0, -840, 0, 54600, 5040, 1680;

%e 0,1920, 0, -13440, 0, 524160, 40320, 11520;

%e -3024, 0, 60480, 0, -211680, 0, 5503680, 362880, 90720;

%e 0, -241920, 0, 1612800, 0, -3386880, 0, 62899200, 3628800, 806400;

%e 604800, 0, -11975040, 0, 39916800, 0, -55883520, 0, 778377600, 39916800, 7983360;

%e 0,72576000, 0, -479001600, 0, 958003200, 0, -958003200, 0, 10378368000, 479001600, 87091200;

%t p[t_] = Exp[x*t](t*(1 - Exp[t]) - Exp[t])/(1 - Exp[t]);

%t a = Table[ CoefficientList[(n!*(n + 2)!*2)*SeriesCoefficient[

%t Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];

%t Flatten[a]

%Y Cf. A176989.

%K sign,tabf

%O 0,1

%A _Roger L. Bagula_, Dec 08 2010