A345651 Fourth column of A008296.

1, 10, 25, -35, 49, 0, -820, 9020, -87164, 859144, -8965320, 100136400, -1199838576, 15406135488, -211479420096, 3094582896000, -48129022468224, 793274283938304, -13818265424460288, 253731538514893824, -4899371564756837376, 99261476593521868800

Offset

4,2

Links

Table of n, a(n) for n=4..25.

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,4).

Formula

a(n) = A008296(n,4).

a(n) = (-1)^n*(4*H(n-5,1)^3 + 8*H(n-5,3) - 12*H(n-5,2)*H(n-5,1) - 25*H(n-5,1)^2 + 25*H(n-5,2) + 35*H(n-5,1) - 10)*(n-5)! for n >= 5 where H(n,1) = Sum_{j=1..n} 1/j is the n-th harmonic number, H(n,2) = Sum_{j=1..n} 1/j^2 and H(n,3) = Sum_{j=1..n} 1/j^3.

a(n) = Sum_{m=4..n} binomial(m,4) * 4^(m-4) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021

Conjecture: D-finite with recurrence a(n) +2*(2*n-13)*a(n-1) +(6*n^2-84*n+295)*a(n-2) +(2*n-15)*(2*n^2-30*n+113)*a(n-3) +(n-8)^4*a(n-4)=0. - R. J. Mathar, Sep 15 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, 4):
seq(a(n), n=4..28);  # Alois P. Heinz, Aug 26 2021
# alternative
seq(A008296(n, 4), n=4..70) ; # R. J. Mathar, Sep 15 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[N[a[n + 4, 4], 10], {n, 1, 400}]]

Prog

(PARI) a(n) = sum(m=4, n, binomial(m, 4)*4^(m-4)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021

Crossrefs

Cf. A008296, A045406, A081048.

Cf. A001008, A002805, A007406, A007407.

Sequence in context: A174051 A274046 A014090 * A154057 A074814 A002600

Adjacent sequences: A345648 A345649 A345650 * A345652 A345653 A345654

Keyword

sign

Author

Luca Onnis, Aug 26 2021

Status

approved