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 A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx). 2
 1, 1, 1, 2, 3, 2, 6, 11, 12, 6, 24, 50, 70, 60, 24, 120, 274, 450, 510, 360, 120, 720, 1764, 3248, 4410, 4200, 2520, 720, 5040, 13068, 26264, 40614, 47040, 38640, 20160, 5040, 40320, 109584, 236248, 403704, 538776, 544320, 393120, 181440, 40320 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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 Edgar remarks that these coefficients are related to Stirling numbers of the second kind (cf. A008277). 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 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 LINKS Nathaniel Johnston, Table of n, a(n) for n = 1..2500 G. A. Edgar, Transseries for beginners, arXiv:0801.4877 [math.RA], 2008-2009. F. Qi, On multivariate logarithmic polynomials and their properties, Indagationes Mathematicae (2018). Wikipedia, Stirling numbers of the first kind FORMULA T(n, k) = (k-1)!*Sum_{i=0..k}(Stirling2(i,k)*(-1)^(n-i)*Stirling1(n,i)) = 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. T(n,k) = [x^k] n!*[t^n](-log(1 + x*log(1 - t))). - Peter Luschny, Jul 10 2020 EXAMPLE Triangle begins: 1 1    1 2    3    2 6    11   12   6 24   50   70   60   24 120  274  450  510  360  120 ... MAPLE 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: T:=proc(n, k)return (k-1)!*S(n, k); end: 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 # With offset n = 0, k = 0: A188881 := (n, k) -> k!*abs(Stirling1(n+1, k+1)): seq(seq(A188881(n, k), k=0..n), n=0..8); # Peter Luschny, Oct 19 2017 # Alternative: gf := -log(1 + x*log(1 - t)): ser := series(gf, t, 18): toeff := n -> n!*expand(coeff(ser, t, n)): seq(print(seq(coeff(toeff(n), x, k), k=1..n)), n=1..8); # Peter Luschny, Jul 10 2020 MATHEMATICA 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 *) PROG (Maxima) T(n, k):=(k-1)!*sum(stirling2(i, k)*(-1)^(n-i)*stirling1(n, i), i, 0, k); /* Vladimir Kruchinin, Apr 17 2015 */ (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 */ (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 */ CROSSREFS Cf. A277408, A003713, A263634. Sequence in context: A087454 A059446 A298854 * A143806 A276551 A109878 Adjacent sequences:  A188878 A188879 A188880 * A188882 A188883 A188884 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Apr 14 2011 EXTENSIONS a(11)-a(45) from Nathaniel Johnston, Apr 15 2011 STATUS approved

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Last modified January 19 02:30 EST 2022. Contains 350464 sequences. (Running on oeis4.)