The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A277409 a(n) equals the coefficient of x^n in (1 - log(1-x))^n! for n>=0. 1

%I #13 Oct 30 2016 08:39:37

%S 1,1,2,37,13921,207504608,193499235977786,16390183551007874514674,

%T 173238206541606827885872411575542,

%U 300679807333480520851459179939426369369129736,109110688416565628491410454990885244124132946665282604804584,10269686361506102165964632192322962717141565478713927846953403915348531319392,304583662721691547994723721287871614789227410136168948343531184046989057630321931742841867554016

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

%F a(n) = Sum_{k=0..n} binomial(n!,k) * k!/n! * (-1)^(n-k) * Stirling1(n,k).

%e Illustration of initial terms.

%e a(0) = 1;

%e a(1) = [x^1] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^1! = 1 ;

%e a(2) = [x^2] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^2!, or

%e a(2) = [x^2] (1 + 2*x + 2*x^2 + 5/3*x^3 + 17/12*x^4 +...) = 2 ;

%e a(3) = [x^3] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^3!, or

%e a(3) = [x^3] (1 + 6*x + 18*x^2 + 37*x^3 + 241/4*x^4 +...) = 37 ;

%e a(4) = [x^4] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^4!, or

%e a(4) = [x^4] (1 + 24*x + 288*x^2 + 2308*x^3 + 13921*x^4 +...) = 13921 ;

%e ...

%e a(n) = [x^n] (1 + x + x^2/2 + x^3/3 + x^4/4 +...+ x^k/k +...)^n! ;

%e ...

%e The coefficients of x^k, k=0..n, in (1 - log(1-x))^n! forms the triangle T(n,k):

%e [1];

%e [1, 1];

%e [1, 2, 2];

%e [1, 6, 18, 37];

%e [1, 24, 288, 2308, 13921];

%e [1, 120, 7200, 288020, 8642405, 207504608];

%e [1, 720, 259200, 62208120, 11197526430, 1612462485648, 193499235977786];

%e [1, 5040, 12700800, 21337344840, 26885057673810, 27100144537250736, 22764130374754974422, 16390183551007874514674];

%e [1, 40320, 812851200, 10924720134720, 110121179161192080, 888017192033323164288, 5967475567171901800336816, 34372659584069639646227206672, 173238206541606827885872411575542]; ...

%e in which the main diagonal forms this sequence: a(n) = T(n,n),

%e where

%e T(n,k) = Sum_{j=0..k} binomial(n!, j) * j!/k! * (-1)^(k-j) * Stirling1(k, j).

%o (PARI) {a(n) = polcoeff( (1 - log(1-x +x*O(x^n)))^n!, n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n) = sum(k=0,n, binomial(n!,k) * k!/n! * (-1)^(n-k) * stirling(n,k,1) )}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {T(n,k) = sum(j=0,k, binomial(n!, j) * j!/k! * (-1)^(k-j) * stirling(k, j, 1) )}

%o for(n=0,20,print1(T(n,n),", "))

%Y Cf. A277759.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 28 2016

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 8 07:12 EST 2023. Contains 367662 sequences. (Running on oeis4.)