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A277409 a(n) equals the coefficient of x^n in (1 - log(1-x))^n! for n>=0.

%I A277409 #13 Oct 30 2016 08:39:37
%S A277409 1,1,2,37,13921,207504608,193499235977786,16390183551007874514674,
%T A277409 173238206541606827885872411575542,
%U A277409 300679807333480520851459179939426369369129736,109110688416565628491410454990885244124132946665282604804584,10269686361506102165964632192322962717141565478713927846953403915348531319392,304583662721691547994723721287871614789227410136168948343531184046989057630321931742841867554016
%N A277409 a(n) equals the coefficient of x^n in (1 - log(1-x))^n! for n>=0.
%F A277409 a(n) = Sum_{k=0..n} binomial(n!,k) * k!/n! * (-1)^(n-k) * Stirling1(n,k).
%e A277409 Illustration of initial terms.
%e A277409 a(0) = 1;
%e A277409 a(1) = [x^1] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^1! = 1 ;
%e A277409 a(2) = [x^2] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^2!, or
%e A277409 a(2) = [x^2] (1 + 2*x + 2*x^2 + 5/3*x^3 + 17/12*x^4 +...) = 2 ;
%e A277409 a(3) = [x^3] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^3!, or
%e A277409 a(3) = [x^3] (1 + 6*x + 18*x^2 + 37*x^3 + 241/4*x^4 +...) = 37 ;
%e A277409 a(4) = [x^4] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^4!, or
%e A277409 a(4) = [x^4] (1 + 24*x + 288*x^2 + 2308*x^3 + 13921*x^4 +...) = 13921 ;
%e A277409 ...
%e A277409 a(n) = [x^n] (1 + x + x^2/2 + x^3/3 + x^4/4 +...+ x^k/k +...)^n! ;
%e A277409 ...
%e A277409 The coefficients of x^k, k=0..n, in (1 - log(1-x))^n! forms the triangle T(n,k):
%e A277409 [1];
%e A277409 [1, 1];
%e A277409 [1, 2, 2];
%e A277409 [1, 6, 18, 37];
%e A277409 [1, 24, 288, 2308, 13921];
%e A277409 [1, 120, 7200, 288020, 8642405, 207504608];
%e A277409 [1, 720, 259200, 62208120, 11197526430, 1612462485648, 193499235977786];
%e A277409 [1, 5040, 12700800, 21337344840, 26885057673810, 27100144537250736, 22764130374754974422, 16390183551007874514674];
%e A277409 [1, 40320, 812851200, 10924720134720, 110121179161192080, 888017192033323164288, 5967475567171901800336816, 34372659584069639646227206672, 173238206541606827885872411575542]; ...
%e A277409 in which the main diagonal forms this sequence: a(n) = T(n,n),
%e A277409 where
%e A277409 T(n,k) = Sum_{j=0..k} binomial(n!, j) * j!/k! * (-1)^(k-j) * Stirling1(k, j).
%o A277409 (PARI) {a(n) = polcoeff( (1 - log(1-x +x*O(x^n)))^n!, n)}
%o A277409 for(n=0,20,print1(a(n),", "))
%o A277409 (PARI) {a(n) = sum(k=0,n, binomial(n!,k) * k!/n! * (-1)^(n-k) * stirling(n,k,1) )}
%o A277409 for(n=0,20,print1(a(n),", "))
%o A277409 (PARI) {T(n,k) = sum(j=0,k, binomial(n!, j) * j!/k! * (-1)^(k-j) * stirling(k, j, 1) )}
%o A277409 for(n=0,20,print1(T(n,n),", "))
%Y A277409 Cf. A277759.
%K A277409 nonn
%O A277409 0,3
%A A277409 _Paul D. Hanna_, Oct 28 2016