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

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A303638 Coefficients of a representation of gamma_{n-1}(1) - gamma_{n-1}(n) where gamma_n(x) are the generalized Euler-Stieltjes constants, triangle read by rows, for n >= 1 and 0 <= k <= n-1. 1

%I #20 Jul 22 2019 05:45:13

%S 1,2,0,6,0,3,24,0,12,8,120,0,540,40,0,720,0,6120,240,0,144,5040,0,

%T 83160,1680,0,1008,840,40320,0,1310400,13440,0,8064,6720,5760,362880,

%U 0,321012720,120960,0,72576,60480,51840,0,3628800,0,9394509600,207648000,0,725760,604800,518400,0,0

%N Coefficients of a representation of gamma_{n-1}(1) - gamma_{n-1}(n) where gamma_n(x) are the generalized Euler-Stieltjes constants, triangle read by rows, for n >= 1 and 0 <= k <= n-1.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Stieltjes_constants#Generalized_Stieltjes_constants">Generalized Stieltjes constants</a>

%F gamma_{n-1}(1) - gamma_{n-1}(n) = (1/n!) Sum_{k=1..n-1} T(n,k)*(log(k))^(n-1) where T(n, k) = 0 if k is a prime power (in the sense of A025475).

%F -Gamma(n)*B^(n)(0,n) = n!*gamma_{n-1} - Sum_{k=1..n-1} T(n,k)(log(k))^(n-1) where Gamma(n) is Euler's Gamma function and B^(n)(0,n) is the n-th derivative of the generalized Bernoulli function B(s, a) with respect to s.

%F Four cases can be distinguished:

%F (1) If k=0 then T(n, k) = n!,

%F (2) else if k is prime then T(n, k) = Sum_{v=1..m} v^(n-1)*k^(-v) where m = ilog_k(n-1) and ilog is the integer base k logarithm,

%F (3) else if k is a prime power in the sense of A025475 then T(n, k) = 0,

%F (4) else (k is composite but not a prime power) T(n, k) = n!/k.

%e The triangle starts:

%e [n\k][ 0 1 2 3 4 5 6 7 8 9]

%e [ 1] [ 1]

%e [ 2] [ 2, 0]

%e [ 3] [ 6, 0, 3]

%e [ 4] [ 24, 0, 12, 8]

%e [ 5] [ 120, 0, 540, 40, 0]

%e [ 6] [ 720, 0, 6120, 240, 0, 144]

%e [ 7] [ 5040, 0, 83160, 1680, 0, 1008, 840]

%e [ 8] [ 40320, 0, 1310400, 13440, 0, 8064, 6720, 5760]

%e [ 9] [ 362880, 0, 321012720, 120960, 0, 72576, 60480, 51840, 0]

%e [10] [3628800, 0, 9394509600, 207648000, 0, 725760, 604800, 518400, 0, 0]

%p Trow := proc(n) local h, r, e, f;

%p h := (n, k) -> `if`(k = 1, x[0], h(n, k-1) - log(k-1)^n/(k-1));

%p r := `if`(n = 0, 1, n!*h(n-1,n)); f := k -> (-x[k])^(1/(n-1));

%p e := eval(subs(ln = f, r)); seq(coeff(e, x[i]), i=0..n-1) end:

%p seq(Trow(n), n=1..10);

%p # Alternative:

%p T := proc(n, k) local ispp, omega:

%p omega := n -> nops(numtheory:-factorset(n)):

%p ispp := n -> not isprime(n) and omega(n) = 1:

%p if k = 0 then return n! fi;

%p if isprime(k) then

%p add(v^(n-1)*k^(-v), v=1..ilog[k](n-1)):

%p return n!*% fi:

%p if k = 1 or ispp(k) then return 0 fi:

%p return n!/k end:

%p seq(seq(T(n,k), k=(0..n-1)), n=1..10);

%t T[n_, k_] := Module[{s}, If[k == 0, Return[n!]]; If[PrimeQ[k], s = Sum[v^(n-1) k^(-v), {v, 1, Log[k, n-1]}]; Return[n! s]]; If[k == 1 || PrimePowerQ[k], Return[0]]; n!/k];

%t Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jul 22 2019, from 2nd Maple program *)

%Y See the cross-references in A301816 for the values of some Stieltjes constants.

%Y Row sums are A303938.

%K nonn,tabl

%O 1,2

%A _Peter Luschny_, Apr 27 2018

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)