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, 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
Offset: 1
Examples
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]
Links
- Wikipedia, Generalized Stieltjes constants
Crossrefs
Programs
-
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-François Alcover, Jul 22 2019, from 2nd Maple program *)
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.