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A176665 Triangle of polynomial coefficients of p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k), read by rows.

%I A176665 #6 Feb 07 2021 19:42:39
%S A176665 1,1,2,1,-5,9,1,109,-165,64,1,-3303,6188,-3494,625,1,169711,-357254,
%T A176665 254434,-74635,7776,1,-13084359,30063342,-24927719,9549230,-1718079,
%U A176665 117649,1,1417404703,-3486909736,3229823067,-1474126800,354928391,-43216649,2097152
%N A176665 Triangle of polynomial coefficients of p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k), read by rows.
%C A176665 Row sums are: A083318 = {1, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, ...}.
%H A176665 G. C. Greubel, <a href="/A176665/b176665.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A176665 Let p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k) then the number triangle is given by T(n, m) = coefficients( p(x,n) ).
%e A176665 Triangle begins as:
%e A176665   1;
%e A176665   1,         2;
%e A176665   1,        -5,        9;
%e A176665   1,       109,     -165,        64;
%e A176665   1,     -3303,     6188,     -3494,     625;
%e A176665   1,    169711,  -357254,    254434,  -74635,     7776;
%e A176665   1, -13084359, 30063342, -24927719, 9549230, -1718079, 117649;
%t A176665 (* First program *)
%t A176665 p[x_, n_]:= Sum[(k+1)^n*k!*Binomial[x, k], {k, 0, n}];
%t A176665 Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]//Flatten
%t A176665 (* Second program *)
%t A176665 f[n_]:= CoefficientList[Sum[(k+1)^n*Product[x-j, {j,0,k-1}], {k,0,n}], x];
%t A176665 Table[f[n], {n, 0, 10}] (* _G. C. Greubel_, Feb 07 2021 *)
%o A176665 (Sage)
%o A176665 def p(n, x): return sum( (k+1)^n*factorial(k)*binomial(x, k) for k in (0..n))
%o A176665 flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Feb 07 2021
%Y A176665 Cf. A083318.
%K A176665 sign,tabl
%O A176665 0,3
%A A176665 _Roger L. Bagula_, Apr 23 2010
%E A176665 Edited by _G. C. Greubel_, Feb 07 2021