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A347276
Third column of A008296.
2
1, 6, 5, -15, 49, -196, 944, -5340, 34716, -254760, 2078856, -18620784, 180973584, -1887504768, 20887922304, -242111586816, 2889841121280, -34586897978880, 393722260047360, -3659128846433280, 5687630494110720, 1137542166526464000, -49644151627682304000
OFFSET
3,2
LINKS
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,3).
FORMULA
a(n) = A008296(n,3).
a(n) = (-1)^n*(3*H(n-4,1)^2 - 3*H(n-4,2) - 11*H(n-4,1) + 6)*(n-4)! for n >= 4, where H(n,1) = Sum_{j=1..n} 1/j = A001008(n)/A002805(n) is the n-th harmonic number and H(n,2) = Sum_{j=1..n} 1/j^2 = A007406(n)/A007407(n).
a(n) = Sum_{m=3..n} binomial(m,3) * 3^(m-3) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021
MAPLE
b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
(n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
end:
a:= n-> b(n, 3):
seq(a(n), n=3..30); # Alois P. Heinz, Aug 25 2021
MATHEMATICA
a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
a[n_, n_] = 1;
a[n_, k_] := a[n, k] = (n - 1) a[n - 2, k - 1] +
a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
Flatten[Table[a[n + 3, 3], {n, 1, 400}]]]]
PROG
(PARI) a(n) = sum(m=3, n, binomial(m, 3)*3^(m-3)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021
CROSSREFS
KEYWORD
sign
AUTHOR
Luca Onnis, Aug 25 2021
STATUS
approved