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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

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.

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)) =

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.

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

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.

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 July 6 23:01 EDT 2020. Contains 335484 sequences. (Running on oeis4.)