A132958 -id:A132958 - cp's OEIS frontend

A132960 a(n) = n!Sum_{d|n} (-1)^(d+1)/(d!(n/d)^d).

1, 0, 3, 2, 25, 94, 721, 3674, 42561, 291248, 3628801, 34254604, 479001601, 5337581534, 88966701825, 1140807642974, 20922789888001, 321094542593824, 6402373705728001, 109338195253235948, 2457732174030848001

Offset: 1

Vladeta Jovovic, Sep 06 2007

G. C. Greubel, Table of n, a(n) for n = 1..450

Cf. A005225, A132958, A132959, A132961, A132962, A132963.

Rest[ Range[0, 22]! CoefficientList[ Series[ - Sum[ Exp[ -x^k/k], {k, 25}], {x, 0, 22}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

a(n) = n!*sumdiv(n, d, (-1)^(d+1)/(d!*(n/d)^d)); \\ Michel Marcus, Sep 29 2017

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

More terms from Robert G. Wilson v, Sep 13 2007

A132962 a(n) = n!Sum_{d|n} (-1)^(d+1)/(d!(n/d)!^d).

Original entry on oeis.org

1, 0, 2, -3, 2, 5, 2, -140, 282, 819, 2, -20482, 2, 133419, 1527528, -4661085, 2, -153296429, 2, 1402482796, 36278688162, 13748957859, 2, -14081800718427, 5194672859378, 7905848380325, 2977584150505252, 12956452725792600, 2, -1314647260913859151

Offset: 1

Vladeta Jovovic, Sep 06 2007

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A038041, A132958, A132959, A132960, A132961, A132963.

Rest[ Range[0, 30]! CoefficientList[ Series[ Sum[1 - Exp[ -x^k/k! ], {k, 30}], {x, 0, 30}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

a(n) = n!*sumdiv(n, d, (-1)^(d+1)/(d!*(n/d)!^d)); \\ Michel Marcus, Sep 29 2017

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

More terms from Robert G. Wilson v, Sep 13 2007

A132959 Total number of all distinct list sizes in all partitions of [n] into lists, cf. A000262.

Original entry on oeis.org

1, 3, 19, 109, 881, 7621, 77785, 854225, 10750465, 143737381, 2121714761, 33426065905, 568250246305, 10242445089605, 197388381934801, 4003553262384961, 86010508861504385, 1939950117886565125

Offset: 1

Vladeta Jovovic, Sep 06 2007

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf. A052852, A132958, A132960, A132961, A132962, A132963.

Rest[ Range[0, 20]! CoefficientList[ Series[ Exp[x/(1 - x)] Sum[(-x)^k/(k!*(x^k - 1)), {k, 25}], {x, 0, 20}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

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

More terms from Robert G. Wilson v, Sep 13 2007

A132961 Total number of all distinct cycle sizes in all permutations of [n].

Original entry on oeis.org

1, 2, 9, 38, 215, 1384, 10409, 86946, 825075, 8541998, 97590779, 1205343952, 16148472977, 231416203212, 3560209750005, 58104163643054, 1008693571819919, 18477578835352366, 357476371577422955, 7258865626801695048, 154893910336866444009, 3454112338490001478772

Offset: 1

Vladeta Jovovic, Sep 06 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..450

Cf. A000254, A132958, A132959, A132960, A132962, A132963.

Row sums of A293211.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j*(p-> p+
      [0, p[1]*`if`(j>0, 1, 0)])(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 21 2015

Rest[ Range[0, 22]! CoefficientList[ Series[1/(1 - x) Sum[1 - Exp[ -x^k/k], {k, 25}], {x, 0, 22}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

E.g.f.: 1/(1-x)*Sum_{k>0} (1-exp(-x^k/k)). Exponential convolution of A132960(n) and n!: a(n) = n!*Sum_{k=1..n} A132960(k)/k!.

More terms from Robert G. Wilson v, Sep 13 2007

A132963 Total number of distinct block sizes in all partitions of [n].

Original entry on oeis.org

1, 2, 8, 25, 102, 439, 2067, 10406, 56754, 328257, 2015818, 13067366, 89192170, 638321285, 4779442602, 37332643831, 303635437532, 2565592977205, 22483754207839, 204013083946460, 1913880812797792, 18536832515581167, 185130415180288134, 1904280138346826637

Offset: 1

Vladeta Jovovic, Sep 06 2007

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A005493, A132958, A132959, A132960, A132961, A132962, A350175.

b:= proc(n, i, c) option remember; `if`(n=0, c,
      `if`(i<1, 0, add(b(n-j*i, i-1, c+signum(j))*
      combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 06 2022

Rest[ Range[0, 23]! CoefficientList[ Series[ Exp[ Exp[x] - 1] Sum[1 - Exp[ -x^k/k! ], {k, 30}], {x, 0, 23}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

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

More terms from Robert G. Wilson v, Sep 13 2007

A330254 Expansion of e.g.f. Sum_{k>=1} sinh(x^k).

Original entry on oeis.org

1, 2, 7, 24, 121, 840, 5041, 40320, 423361, 3659040, 39916801, 558835200, 6227020801, 87195588480, 1536517382401, 20922789888000, 355687428096001, 7469453633241600, 121645100408832001, 2453176191578112000, 59616236292028416001

Offset: 1

Ilya Gutkovskiy, Dec 07 2019

nonn

Cf. A057625, A132958, A176473, A330255.

nmax = 21; CoefficientList[Series[Sum[Sinh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! DivisorSum[n, 1/#! &, OddQ[#] &], {n, 1, 21}]

E.g.f.: Sum_{k>=1} x^(2*k - 1) / ((2*k - 1)! * (1 - x^(2*k - 1))).

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

A330255 Expansion of e.g.f. Sum_{k>=1} (cosh(x^k) - 1) (even powers only).

Original entry on oeis.org

1, 13, 361, 21841, 1814401, 260124481, 43589145601, 11333696774401, 3210079038566401, 1317822591538252801, 562000363888803840001, 336953340897297630105601, 201645730563302817792000001, 165147853334842304408401920001, 132994909752412012763531673600001

Offset: 1

Ilya Gutkovskiy, Dec 07 2019

nonn

Cf. A057625, A132958, A176474, A330254.

nmax = 15; Table[(CoefficientList[Series[Sum[Cosh[x^k] - 1, {k, 1, nmax}], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}] // Rest
Table[(2 n)! DivisorSum[n, 1/(2 #)! &], {n, 1, 15}]

E.g.f.: Sum_{k>=1} x^(2*k) / ((2*k)! * (1 - x^(2*k))) (even powers only).

a(n) = (2*n)! * Sum_{d|n} 1 / (2*d)!.

A182928 Triangular array read by rows: [T(n,k),k=1..tau(n)] = [-n!/(d*(-(n/d)!)^d), d|n].

Original entry on oeis.org

1, 1, -1, 1, 2, 1, -3, -6, 1, 24, 1, -10, 30, -120, 1, 720, 1, -35, -630, -5040, 1, 560, 40320, 1, -126, 22680, -362880, 1, 3628800, 1, -462, 11550, -92400, -1247400, -39916800, 1, 479001600, 1, -1716, 97297200, -6227020800

Offset: 1

Peter Luschny, Apr 13 2011

The number of terms in the n-th row is the number of divisors of n. The n-th row is (apart from sign) a subsequence of the column labeled "M_1" for n-1 in Abramowitz and Stegun, Handbook, p. 831.
Let s(n) be the sum of row n. The number of partitions of an n-set with distinct block sizes can be computed recursively as A007837(0) = 1 and A007837(n) = - Sum_{1<=k<=n} binomial(n-1,k-1)*s(k)*A007837(n-k).
Let t(n) be the sum of the absolute values of row n. The sum of multinomial coefficients can be computed recursively as A005651(0) = 1 and A005651(n) = Sum_{1<=k<=n} binomial(n-1,k-1)*t(k)*A005651(n-k).

			The array starts with
[1] 1,
[2] 1,  -1,
[3] 1,   2,
[4] 1,  -3,   -6,
[5] 1,  24,
[6] 1, -10,   30,  -120,
[7] 1, 720,
[8] 1, -35,  -630, -5040,
[9] 1, 560, 40320,

Cf. A076901, A132958, A132959, A132960, A132962.

A182928_row := proc(n) local d;
seq(-n!/(d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:

row[n_] := Table[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

A327243 a(n) = n! * Sum_{d|n} (-1)^(n - d) / (n/d)!.

Original entry on oeis.org

1, 1, 7, 35, 121, 479, 5041, 62159, 423361, 1844639, 39916801, 779042879, 6227020801, 43606442879, 1536517382401, 32256486662399, 355687428096001, 4259374594675199, 121645100408832001, 3568256949101644799, 59616236292028416001, 562000392047391897599

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

nonn

Cf. A057625, A132958.

[Factorial(n)*(&+[(-1)^(n-d)/Factorial(n div d):d in Divisors(n)]):n in [1..22]]; // Marius A. Burtea, Sep 14 2019

a[n_] := n! Sum[(-1)^(n - d)/(n/d)!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[-(-x)^k/(k! (1 + (-x)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = {n!*sumdiv(n, d, (-1)^(n - d) / (n/d)!)} \\ Andrew Howroyd, Sep 14 2019

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

E.g.f.: Sum_{k>=1} (-1)^k * (exp((-x)^k) - 1). [corrected by Ilya Gutkovskiy, May 14 2022]

A332466 a(n) = n! * Sum_{d|n} mu(d) / d!.

Original entry on oeis.org

1, 1, 5, 12, 119, 241, 5039, 20160, 302400, 1784161, 39916799, 160332480, 6227020799, 43571848321, 1078831353601, 10461394944000, 355687428095999, 2143016754278400, 121645100408831999, 1196177491129420800, 42565648051390464001, 562000335730215782401

Offset: 1

Ilya Gutkovskiy, Feb 13 2020

nonn

Cf. A008683, A057625, A068107, A099740, A132958, A332467.

with(numtheory):
a:= n-> n! * add(mobius(d)/d!, d=divisors(n)):
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 13 2020

Table[n! DivisorSum[n, MoebiusMu[#]/#! &], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] x^k/(k! (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n)={sumdiv(n, d, moebius(d)*n!/d!)} \\ Andrew Howroyd, Feb 13 2020

E.g.f.: Sum_{k>=1} Sum_{j>=1} mu(j) * x^(k*j) / j!.

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

cp's OEIS Frontend

A132960 a(n) = n!*Sum_{d|n} (-1)^(d+1)/(d!*(n/d)^d).

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A132962 a(n) = n!*Sum_{d|n} (-1)^(d+1)/(d!*(n/d)!^d).

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A132959 Total number of all distinct list sizes in all partitions of [n] into lists, cf. A000262.

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A132961 Total number of all distinct cycle sizes in all permutations of [n].

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A132963 Total number of distinct block sizes in all partitions of [n].

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Maple

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A330254 Expansion of e.g.f. Sum_{k>=1} sinh(x^k).

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A330255 Expansion of e.g.f. Sum_{k>=1} (cosh(x^k) - 1) (even powers only).

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A182928 Triangular array read by rows: [T(n,k),k=1..tau(n)] = [-n!/(d*(-(n/d)!)^d), d|n].

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A132960 a(n) = n!Sum_{d|n} (-1)^(d+1)/(d!(n/d)^d).

A132962 a(n) = n!Sum_{d|n} (-1)^(d+1)/(d!(n/d)!^d).