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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.

%I A303638 #20 Jul 22 2019 05:45:13
%S A303638 1,2,0,6,0,3,24,0,12,8,120,0,540,40,0,720,0,6120,240,0,144,5040,0,
%T A303638 83160,1680,0,1008,840,40320,0,1310400,13440,0,8064,6720,5760,362880,
%U A303638 0,321012720,120960,0,72576,60480,51840,0,3628800,0,9394509600,207648000,0,725760,604800,518400,0,0
%N 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.
%H A303638 Wikipedia, <a href="https://en.wikipedia.org/wiki/Stieltjes_constants#Generalized_Stieltjes_constants">Generalized Stieltjes constants</a>
%F A303638 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 A303638 -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 A303638 Four cases can be distinguished:
%F A303638 (1) If k=0 then T(n, k) = n!,
%F A303638 (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 A303638 (3) else if k is a prime power in the sense of A025475 then T(n, k) = 0,
%F A303638 (4) else (k is composite but not a prime power) T(n, k) = n!/k.
%e A303638 The triangle starts:
%e A303638 [n\k][      0  1           2          3  4       5       6       7  8  9]
%e A303638 [ 1] [      1]
%e A303638 [ 2] [      2, 0]
%e A303638 [ 3] [      6, 0,          3]
%e A303638 [ 4] [     24, 0,         12,         8]
%e A303638 [ 5] [    120, 0,        540,        40, 0]
%e A303638 [ 6] [    720, 0,       6120,       240, 0,    144]
%e A303638 [ 7] [   5040, 0,      83160,      1680, 0,   1008,    840]
%e A303638 [ 8] [  40320, 0,    1310400,     13440, 0,   8064,   6720,   5760]
%e A303638 [ 9] [ 362880, 0,  321012720,    120960, 0,  72576,  60480,  51840, 0]
%e A303638 [10] [3628800, 0, 9394509600, 207648000, 0, 725760, 604800, 518400, 0, 0]
%p A303638 Trow := proc(n) local h, r, e, f;
%p A303638 h := (n, k) -> `if`(k = 1, x[0], h(n, k-1) - log(k-1)^n/(k-1));
%p A303638 r := `if`(n = 0, 1, n!*h(n-1,n)); f := k -> (-x[k])^(1/(n-1));
%p A303638 e := eval(subs(ln = f, r)); seq(coeff(e, x[i]), i=0..n-1) end:
%p A303638 seq(Trow(n), n=1..10);
%p A303638 # Alternative:
%p A303638 T := proc(n, k) local ispp, omega:
%p A303638   omega := n -> nops(numtheory:-factorset(n)):
%p A303638   ispp  := n -> not isprime(n) and omega(n) = 1:
%p A303638   if k = 0 then return n! fi;
%p A303638   if isprime(k) then
%p A303638      add(v^(n-1)*k^(-v), v=1..ilog[k](n-1)):
%p A303638      return n!*% fi:
%p A303638   if k = 1 or ispp(k) then return 0 fi:
%p A303638   return n!/k end:
%p A303638 seq(seq(T(n,k), k=(0..n-1)), n=1..10);
%t A303638 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 A303638 Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jul 22 2019, from 2nd Maple program *)
%Y A303638 See the cross-references in A301816 for the values of some Stieltjes constants.
%Y A303638 Row sums are A303938.
%K A303638 nonn,tabl
%O A303638 1,2
%A A303638 _Peter Luschny_, Apr 27 2018