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A176989
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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.
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2
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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 + ...
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EXAMPLE
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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;
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MAPLE
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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:
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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