A355991 -id:A355991

A355989

a(n) = n! / (2 * floor(n/2)!).

+10
5

1, 3, 6, 30, 60, 420, 840, 7560, 15120, 166320, 332640, 4324320, 8648640, 129729600, 259459200, 4410806400, 8821612800, 167610643200, 335221286400, 7039647014400, 14079294028800, 323823762662400, 647647525324800, 16191188133120000, 32382376266240000

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

LINKS

Table of n, a(n) for n=2..26.

FORMULA

E.g.f.: (1 - x^2) * (exp(x^2) - 1)/(2 * (1 - x)).

a(n) = A081125(n)/2.

From Amiram Eldar, Jul 26 2022: (Start)

Sum_{n>=2} 1/a(n) = 3*exp(1/4)*sqrt(Pi)*erf(1/2) - 2, where erf is the error function.

Sum_{n>=2} (-1)^n/a(n) = 2 - exp(1/4)*sqrt(Pi)*erf(1/2). (End)

MATHEMATICA

a[n_] := n!/(2 * Floor[n/2]!); Array[a, 25, 2] (* Amiram Eldar, Jul 22 2022 *)

PROG

(PARI) a(n) = n!/(2*(n\2)!);

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2)*(exp(x^2)-1)/(2*(1-x))))

(Python)

from math import factorial, floor

def a(n): return int(factorial(n)/(2*factorial(floor(n/2))))

print([a(n) for n in range(2, 30)]) # Javier Rivera Romeu, Jul 22 2022

(Python)

from sympy import rf

def A355989(n): return rf((m:=n+1>>1)+(n+1&1), m)>>1 # Chai Wah Wu, Jul 22 2022

CROSSREFS

Column 2 of A355996.

Cf. A355990, A355991,

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Jul 22 2022

STATUS

approved

A355990

a(n) = n! / (6 * floor(n/3)!).

+10
5

1, 4, 20, 60, 420, 3360, 10080, 100800, 1108800, 3326400, 43243200, 605404800, 1816214400, 29059430400, 494010316800, 1482030950400, 28158588057600, 563171761152000, 1689515283456000, 37169336236032000, 854894733428736000, 2564684200286208000

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

LINKS

Table of n, a(n) for n=3..24.

FORMULA

E.g.f.: (1 - x^3) * (exp(x^3) - 1)/(6 * (1 - x)).

a(n) = A355988(n)/6.

MATHEMATICA

a[n_] := n!/(6 * Floor[n/3]!); Array[a, 22, 3] (* Amiram Eldar, Jul 22 2022 *)

PROG

(PARI) a(n) = n!/(6*(n\3)!);

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)*(exp(x^3)-1)/(6*(1-x))))

CROSSREFS

Column 3 of A355996.

Cf. A355988, A355989, A355991,

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Jul 22 2022

STATUS

approved

A355996

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

+10
4

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 30, 20, 5, 1, 1, 60, 60, 30, 6, 1, 1, 420, 420, 210, 42, 7, 1, 1, 840, 3360, 840, 336, 56, 8, 1, 1, 7560, 10080, 7560, 3024, 504, 72, 9, 1, 1, 15120, 100800, 75600, 15120, 5040, 720, 90, 10, 1, 1, 166320, 1108800, 831600, 166320, 55440, 7920, 990, 110, 11, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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

EXAMPLE

Triangle begins:

1;

1, 1;

1, 3, 1;

1, 6, 4, 1;

1, 30, 20, 5, 1;

1, 60, 60, 30, 6, 1;

1, 420, 420, 210, 42, 7, 1;

1, 840, 3360, 840, 336, 56, 8, 1;

...

MATHEMATICA

T[n_, k_] := n!/(k!*Floor[n/k]!); Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 22 2022 *)

PROG

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

CROSSREFS

Row sums give A355991.

Column k=1..3 give A000012, A355989, A355990.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Jul 22 2022

STATUS

approved

A355987

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

+10
3

1, 3, 13, 61, 421, 2641, 23521, 203281, 2071441, 22407841, 286403041, 3453468481, 51122111041, 759194916481, 12216117513601, 203300293996801, 3811792426041601, 69634723878720001, 1444704854104512001, 29725332567567436801, 658231789483184716801

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..21.

FORMULA

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

a(n) ~ c * n! * n, where c = 0.59962032... - Vaclav Kotesovec, Aug 03 2022

Conjecture: c = Sum_{k>=1} 1/((k+1)!*k) = 2 - exp(1) - A001620 + A091725. - Vaclav Kotesovec, Sep 24 2023

MATHEMATICA

a[n_] := n! * Sum[1/Floor[n/k]!, {k, 1, n}]; Array[a, 21] (* Amiram Eldar, Jul 22 2022 *)

PROG

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

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

CROSSREFS

Cf. A007489, A081125, A345683, A355886, A355988, A355991.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jul 22 2022

STATUS

approved

A356011

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

+10
2

1, 2, 6, 17, 80, 337, 2240, 14681, 117010, 1023941, 10900472, 108881665, 1375544846, 17732140805, 247041590476, 3605768497217, 59990390084690, 977383707751621, 18214603019184800, 337615168055209601, 6763842079452393622, 141262515443311046885

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..22.

FORMULA

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

PROG

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

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

CROSSREFS

Row sums of A356013.

Cf. A345682, A345683, A355991, A356015.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jul 23 2022

STATUS

approved

A356004

a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d! * (k/d)!).

+10
1

1, 4, 14, 64, 322, 2054, 14380, 116722, 1060580, 10636042, 116996464, 1411275650, 18346583452, 256869465610, 3856674412952, 61743633813634, 1049641774831780, 18896533652098442, 359034139389870400, 7182372973523436802, 150833211474559084844

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..21.

FORMULA

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

a(n) = n! * Sum_{k=1..n} A121860(k)/k!.

MATHEMATICA

a[n_] := n! * Sum[DivisorSum[k, 1/(#!*(k/#)!) &], {k, 1, n}]; Array[a, 21] (* Amiram Eldar, Jul 22 2022 *)

PROG

(PARI) a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d!*(k/d)!)));

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

CROSSREFS

Cf. A002627, A121860, A355886, A355991.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jul 22 2022

STATUS

approved