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A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

%I A188881 #59 Jul 14 2025 08:17:24
%S A188881 1,1,1,2,3,2,6,11,12,6,24,50,70,60,24,120,274,450,510,360,120,720,
%T A188881 1764,3248,4410,4200,2520,720,5040,13068,26264,40614,47040,38640,
%U A188881 20160,5040,40320,109584,236248,403704,538776,544320,393120,181440,40320
%N A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).
%C A188881 Also the coefficients of the polynomials which are generated by the exponential generating function -log(1 + x*log(1 - t)). The polynomials might be called 'logarithmic polynomials'. Note also A003713, and A263634 for a different use of this term. See the paper of F. Qi for a related, but different family of polynomials. - _Peter Luschny_, Jul 11 2020
%C A188881 Edgar remarks that these coefficients are related to Stirling numbers of the second kind (cf. A008277).
%C A188881 The first column and the main diagonal are the factorials (A000142). The n-th entry on the first subdiagonal is A001710(n+1). The second column is A000254, the third column is 2*A000399, and the fourth column is 6*A000454. In general, the k-th column is (k-1)!*s(n,k), where s(n,k) is the unsigned Stirling number of the first kind. - _Nathaniel Johnston_, Apr 15 2011
%C A188881 With offset n=0, k=0 : triangle T(n,k), read by rows,given by T(n,k) = k*T(n-1, k-1) + n*T(n-1, k) with T(0, 0) = 1. - _Philippe Deléham_, Oct 04 2011
%H A188881 Nathaniel Johnston, <a href="/A188881/b188881.txt">Table of n, a(n) for n = 1..2500</a>
%H A188881 G. A. Edgar, <a href="http://arxiv.org/abs/0801.4877">Transseries for beginners</a>, arXiv:0801.4877 [math.RA], 2008-2009.
%H A188881 F. Qi, <a href="https://doi.org/10.1016/j.indag.2018.04.002">On multivariate logarithmic polynomials and their properties</a>, Indagationes Mathematicae (2018).
%H A188881 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind">Stirling numbers of the first kind</a>
%F A188881 T(n, k) = (k-1)!*Sum_{i=0..k}(Stirling2(i,k)*(-1)^(n-i)*Stirling1(n,i)) =
%F A188881 T(n, k) = Sum_{i=0..k}(W(i,k)*(-1)^(n-i)*Stirling1(n,i)), where W(n,k) is the Worpitzky triangle A028246. - _Vladimir Kruchinin_, Apr 17 2015.
%F A188881 T(n,k) = [x^k] n!*[t^n](-log(1 + x*log(1 - t))). - _Peter Luschny_, Jul 10 2020
%F A188881 T(n,k) = Sum_{m=0..n-k} abs(Stirling1(n-1,m+k-1))*(k+m-1)!/m!. - _Vladimir Kruchinin_, Jul 14 2025
%e A188881 Triangle begins:
%e A188881 1
%e A188881 1    1
%e A188881 2    3    2
%e A188881 6    11   12   6
%e A188881 24   50   70   60   24
%e A188881 120  274  450  510  360  120
%e A188881 ...
%p A188881 S:=proc(n,k)global s:if(n=0 and k=0)then s[0,0]:=1:elif(n=0 or k=0)then s[n,k]:=0:elif(not type(s[n,k],integer))then s[n,k]:=(n-1)*S(n-1,k)+S(n-1,k-1):fi:return s[n,k]:end:
%p A188881 T:=proc(n,k)return (k-1)!*S(n,k);end:
%p A188881 for n from 1 to 6 do for k from 1 to n do print(T(n,k)):od:od: # _Nathaniel Johnston_, Apr 15 2011
%p A188881 # With offset n = 0, k = 0:
%p A188881 A188881 := (n, k) -> k!*abs(Stirling1(n+1, k+1)):
%p A188881 seq(seq(A188881(n,k), k=0..n), n=0..8); # _Peter Luschny_, Oct 19 2017
%p A188881 # Alternative:
%p A188881 gf := -log(1 + x*log(1 - t)): ser := series(gf, t, 18):
%p A188881 toeff := n -> n!*expand(coeff(ser, t, n)):
%p A188881 seq(print(seq(coeff(toeff(n), x, k), k=1..n)), n=1..8); # _Peter Luschny_, Jul 10 2020
%t A188881 Table[(k - 1)! * Sum[StirlingS2[i, k] * (-1)^(n - i) * StirlingS1[n, i], {i, 0, k}], {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, Apr 17 2015 *)
%o A188881 (Maxima)
%o A188881 T(n,k):=(k-1)!*sum(stirling2(i,k)*(-1)^(n-i)*stirling1(n,i),i,0,k); /* _Vladimir Kruchinin_, Apr 17 2015 */
%o A188881 (PARI) {T(n, k) = if( k<1 || k>n, 0, (n-1)! * polcoeff( (x / (1 - exp(-x * (1 + x * O(x^n)))))^n, n-k))}; /* _Michael Somos_, May 10 2017 */
%o A188881 (PARI) {T(n, k) = if( k<1 || n<0, 0, (k-1)! * sum(i=0, k, stirling(i, k, 2) * (-1)^(n-i) * stirling(n, i, 1)))}; /* _Michael Somos_, May 10 2017 */
%Y A188881 Cf. A277408, A003713, A263634.
%K A188881 nonn,tabl
%O A188881 1,4
%A A188881 _N. J. A. Sloane_, Apr 14 2011
%E A188881 a(11)-a(45) from _Nathaniel Johnston_, Apr 15 2011