A309911 a(n) = Sum_{k=1..n} Stirling2(n,k) * floor(n/k).

1, 3, 7, 25, 71, 360, 1310, 7195, 35740, 213318, 1132154, 8409475, 50344672, 366939569, 2728237607, 21375289293, 159969524749, 1462761108082, 11896122581676, 107011124829787, 1031744001100166, 9684995830526129, 91735916202054984, 1010641832989185386, 10131466944871497886

Offset

1,2

Links

Table of n, a(n) for n=1..25.

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k=1..n} Stirling2(n,k) * x^k/(1 - x^k).

a(n) = Sum_{k=1..n} Sum_{d|k} Stirling2(n,d).

Mathematica

Table[Sum[StirlingS2[n, k] Floor[n/k] , {k, 1, n}], {n, 1, 25}]
Table[SeriesCoefficient[1/(1 - x) Sum[StirlingS2[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 25}]
Table[Sum[Sum[StirlingS2[n, d], {d, Divisors[k]}], {k, 1, n}], {n, 1, 25}]

Prog

(PARI) a(n) = sum(k=1, n, stirling(n, k, 2) * (n\k)); \\ Michel Marcus, Aug 23 2019
(Python)
from fractions import Fraction
from math import comb, factorial
def A309911(n):
    c, j = 0, 1
    while j <= n:
        k = n//j
        m = n//k
        c += k*sum((-i**n if i&1 else i**n)*sum((-1 if a&1 else 1)*Fraction(comb(a, i), factorial(a)) for a in range(max(i, j), m+1)) for i in range(m+1))
        j = m+1
    return int(c) # Chai Wah Wu, May 14 2026

Crossrefs

Cf. A008277, A308037, A308812, A309910.

Sequence in context: A148723 A148724 A332508 * A148725 A148726 A148727

Adjacent sequences: A309908 A309909 A309910 * A309912 A309913 A309914

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 22 2019

Status

approved