A356011 -id:A356011 - cp's OEIS frontend

A356013 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)).

1, 1, 1, 2, 3, 1, 6, 6, 4, 1, 24, 30, 20, 5, 1, 120, 120, 60, 30, 6, 1, 720, 840, 420, 210, 42, 7, 1, 5040, 5040, 3360, 840, 336, 56, 8, 1, 40320, 45360, 20160, 7560, 3024, 504, 72, 9, 1, 362880, 362880, 201600, 75600, 15120, 5040, 720, 90, 10, 1

Offset: 1

Seiichi Manyama, Jul 23 2022

			Triangle begins:
      1;
      1,     1;
      2,     3,     1;
      6,     6,     4,    1;
     24,    30,    20,    5,    1;
    120,   120,    60,   30,    6,   1;
    720,   840,   420,  210,   42,   7,  1;
   5040,  5040,  3360,  840,  336,  56,  8, 1;
  40320, 45360, 20160, 7560, 3024, 504, 72, 9, 1;
  ...

Row sums gives A356011.
Column k=1..3 give A000142(n-1), |A265376(n)|, A356012.
Cf. A355996.

PARI
```
T(n, k) = n!/(k!*(n\k));
```

E.g.f. of column k: -(1 - x^k) * log(1 - x^k)/(k! * (1 - x)).

A356015 a(n) = n! * Sum_{k=1..n} 1/(k * floor(n/k)!).

Original entry on oeis.org

1, 2, 6, 21, 125, 625, 5089, 38185, 343657, 3376081, 40765681, 427649761, 6038448481, 84486386881, 1252766088001, 19388604009601, 350529058051201, 5938944734419201, 119242323659692801, 2303746722596390401, 48358406991122726401, 1063884813011759692801

Offset: 1

Seiichi Manyama, Jul 23 2022

nonn

Vaclav Kotesovec, Plot of a(n)/n! for n = 1..10000

Cf. A355987, A356011.

Table[n! * Sum[1 / (k*Floor[n/k]!), {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *)

PARI
```
a(n) = n!*sum(k=1, n, 1/(k*(n\k)!));
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (1-x^k)*(exp(x^k)-1)/k)/(1-x)))

E.g.f.: (1/(1-x)) * Sum_{k>0} (1 - x^k) * (exp(x^k) - 1)/k.

Conjecture: a(n) ~ c * n!, where c = 0.95488757... - Vaclav Kotesovec, Aug 11 2025

cp's OEIS Frontend

A356013 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)).

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A356015 a(n) = n! * Sum_{k=1..n} 1/(k * floor(n/k)!).

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