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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
1, 2, 0, 6, 0, 3, 24, 0, 12, 8, 120, 0, 540, 40, 0, 720, 0, 6120, 240, 0, 144, 5040, 0, 83160, 1680, 0, 1008, 840, 40320, 0, 1310400, 13440, 0, 8064, 6720, 5760, 362880, 0, 321012720, 120960, 0, 72576, 60480, 51840, 0, 3628800, 0, 9394509600, 207648000, 0, 725760, 604800, 518400, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..55.

Wikipedia, Generalized Stieltjes constants

FORMULA

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).

-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.

Four cases can be distinguished:

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

(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,

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

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

EXAMPLE

The triangle starts:

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

[ 1] [      1]

[ 2] [      2, 0]

[ 3] [      6, 0,          3]

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

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

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

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

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

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

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

MAPLE

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

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

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

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

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

# Alternative:

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

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

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

  if k = 0 then return n! fi;

  if isprime(k) then

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

     return n!*% fi:

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

  return n!/k end:

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

MATHEMATICA

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];

Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-Fran├žois Alcover, Jul 22 2019, from 2nd Maple program *)

CROSSREFS

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

Row sums are A303938.

Sequence in context: A050821 A076257 A274881 * A162974 A275325 A300227

Adjacent sequences:  A303635 A303636 A303637 * A303639 A303640 A303641

KEYWORD

nonn,tabl,changed

AUTHOR

Peter Luschny, Apr 27 2018

STATUS

approved

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Last modified July 22 16:34 EDT 2019. Contains 325224 sequences. (Running on oeis4.)