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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

%I #12 May 17 2019 02:35:55

%S 2,2,5,6,3,2,40,24,8,-1,30,300,120,30,-12,-24,360,2400,720,144,20,

%T -420,-420,4200,21000,5040,840,480,960,-10080,-6720,50400,201600,

%U 40320,5760,-1512,30240,30240,-211680,-105840,635040,2116800,362880,45360,-60480,-120960,1209600,806400

%N 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.

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

%C 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.

%F 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 + ...

%e The coefficients start in row n=0 with column k=0..n+1 as:

%e 2, 2;

%e 5, 6, 3;

%e 2, 40, 24, 8;

%e -1, 30, 300, 120, 30;

%e -12, -24, 360, 2400, 720, 144;

%e 20, -420, -420, 4200, 21000, 5040, 840;

%e 480, 960, -10080, -6720, 50400, 201600, 40320, 5760;

%p A176989 := proc(n,k) local x ;

%p exp(x*t)*(t*(1-2*exp(t))-2*exp(t))/(1-exp(t))-2/t ;

%p n!*(n+2)!/2 *% ; series(%,t,n+3) ;

%p convert(%,polynom) ; coeftayl(%,t=0,n) ; coeftayl(%,x=0,k) ;

%p end proc:

%p seq (seq(A176989(n,k),k=0..n+1),n=0..5) ; # _R. J. Mathar_, Dec 20 2010

%t p[t_] = Exp[x*t]*(t*(1 - 2*Exp[t]) - 2*Exp[t])/(1 - Exp[t]);

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

%t Flatten[a]

%K sign,tabf

%O 0,1

%A _Roger L. Bagula_, Dec 08 2010