A024167 -id:A024167 - cp's OEIS frontend

A078921 Signed variant of A077012.

1, -1, 2, 2, -3, 6, -6, 8, -12, 24, 24, -30, 40, -60, 120, -120, 144, -180, 240, -360, 720, 720, -840, 1008, -1260, 1680, -2520, 5040, -5040, 5760, -6720, 8064, -10080, 13440, -20160, 40320, 40320, -45360, 51840, -60480, 72576, -90720, 120960, -181440, 362880, -362880, 403200, -453600, 518400, -604800, 725760, -907200, 1209600, -1814400, 3628800

Offset: 1

Wouter Meeussen, Dec 14 2002

Row sums give A024167.

			Triangle starts:
     1
    -1,    2
     2,   -3,    6
    -6,    8,  -12,    24
    24,  -30,   40,   -60,  120
  -120,  144, -180,   240, -360,   720
   720, -840, 1008, -1260, 1680, -2520, 5040
  ...

Cf. A000254, A024167, A077012, A078922.

Table[Table[ -(-1)^(n-k+1) n/(n-k+1), {k, 1, n}] (n-1)!, {n, 1, 12}]

T(n, k) = -(-1)^(n-k+1)*(n/(n-k+1))*(n-1)!.
E.g.f.: log(1+x)/(1-y*x). - Vladeta Jovovic, Feb 07 2003
Sum_{n>=1} Sum_{k=1..n} 1/T(n, k) = (e^2+1)/(4*e). - Amiram Eldar, Jun 29 2025

A087301 a(n) = n!*Sum_{i=1..n-1} (-1)^(i+1)/i.

Original entry on oeis.org

2, 3, 20, 70, 564, 3108, 30624, 230256, 2705760, 25771680, 352805760, 4067556480, 63651813120, 861371884800, 15176802816000, 235775183616000, 4620563523072000, 81032645804544000, 1748700390205440000

Offset: 2

Vladeta Jovovic, Oct 20 2003

nonn

Stirling transform of A052882(n)=[0,2,9,52,375,...] is a(n+1)=[0,2,3,20,...]. - Michael Somos, Mar 04 2004

Cf. A024167, A052881.

Rest[Table[n!Sum[(-1)^(i+1)/i,{i,n-1}],{n,20}]] (* Harvey P. Dale, Oct 24 2011 *)

a(n)=if(n<0,0,n!*polcoeff(log(1+x+x*O(x^n))*x/(1-x),n))

E.g.f.: x*log(1+x)/(1-x). a(n) = 1/2*(-1)^n*n!*(2*(-1)^n*log(2)+Psi(1/2+1/2*n)-Psi(1/2*n)).

a(n) ~ n! * log(2). - Vaclav Kotesovec, Jul 01 2018

A346943 a(n) = a(n-1) + n(n+1)a(n-2) with a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 7, 19, 159, 729, 7407, 48231, 581535, 4922325, 68891175, 718638075, 11465661375, 142257791025, 2550046679775, 36691916525775, 730304613424575, 11958031070311725, 261722208861516375, 4805774015579971875, 114729101737416849375, 2334996696935363855625

Offset: 0

Vaclav Kotesovec, Aug 08 2021, following a suggestion from John M. Campbell

From Peter Bala, Dec 09 2024: (Start)

b(n) := A000246(n+2) = (n+2)!/2^(n+1) * binomial(n+1, floor((n+1)/2)) satisfies the same second-order recurrence as a(n) with the initial conditions b(0) = 1 and b(1) = 3. This leads to the finite continued fraction a(n)/b(n) = 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/1). Letting n tend to infinity gives the continued fraction representation 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/(1 + ...) = Pi/2 - 1, due to Euler - see paragraph 31, p. 48. (End)

Leonhard Euler, E123: De fractionibus continuis observationes, originally published in Commentarii academiae scientiarum Petropolitanae 11, 1750, pp. 32-81, reprinted in Opera Omnia: Series 1, Volume 14, 291-349.

Cf. A000246, A000932, A024167.

RecurrenceTable[{a[n] == a[n-1] + n*(n+1)*a[n-2], a[0]==1, a[1]==1}, a, {n,0,20}]
nmax = 20; CoefficientList[Series[(-2 + Pi + 2*Pi*x + 4*Sqrt[1 - x^2] + 2*x*(-2 + Sqrt[1 - x^2]) - 4*(1 + 2*x) * ArcSin[Sqrt[1 - x]/Sqrt[2]]) / (2*(1 - x)^(5/2) * (1 + x)^(3/2)), {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! * (Pi - 2) * n^(3/2) / sqrt(2*Pi).
a(n) ~ (Pi - 2) * n^(n+2) / exp(n).
E.g.f. A(x) satisfies the differential equation -6*A(x) - (6*x + 1)*A'(x) + (1 - x^2)*A''(x) = 0, A(0)=1, A'(0)=1.
E.g.f.: (-2 + Pi + 2*Pi*x + 4*sqrt(1-x^2) + 2*x*(-2+sqrt(1-x^2)) - 4*(1+2*x) * arcsin(sqrt(1-x)/sqrt(2))) / (2*(1-x)^(5/2) * (1+x)^(3/2)).

A371685 Triangle read by rows: T(n, k) = n! * Sum_{j=0..n-1} binomial(k - 1, j) / (j + 1).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 6, 9, 14, 14, 24, 36, 56, 90, 94, 120, 180, 280, 450, 744, 444, 720, 1080, 1680, 2700, 4464, 7560, 3828, 5040, 7560, 11760, 18900, 31248, 52920, 91440, 25584, 40320, 60480, 94080, 151200, 249984, 423360, 731520, 1285200

Offset: 0

Peter Luschny, Apr 06 2024

			Triangle starts:
[0]    0;
[1]    1,    1;
[2]    1,    2,    3;
[3]    5,    6,    9,    14;
[4]   14,   24,   36,    56,    90;
[5]   94,  120,  180,   280,   450,   744;
[6]  444,  720, 1080,  1680,  2700,  4464,  7560;
[7] 3828, 5040, 7560, 11760, 18900, 31248, 52920, 91440;

Cf. A029767 (main diagonal), A024167 (column 0), A371768 (row sums).

T := (n, k) -> local j; n!*add(binomial(k-1, j)/(j + 1), j = 0..n-1):
T := (n, k) -> local j; n!*ifelse(n = 0, 0, ifelse(k=0, add(-(-1)^j/j, j = 1..n), (2^k - 1) / k)):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);

Restricted to the range 1 <= k <= n: T(n, k) = n!*(2^k - 1)/k.

A302582 a(n) = n! * [x^n] log(1 + x)/(1 - x)^n.

Original entry on oeis.org

0, 1, 3, 29, 386, 6774, 146484, 3762744, 111868560, 3777096240, 142734788640, 5967788097600, 273488036169600, 13631083378617600, 734083968523046400, 42477063883483622400, 2628184745184816384000, 173147202267665649408000, 12100888735302910523904000, 894183767796064712795136000

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

Cf. A000254, A024167, A058806, A104150, A109792, A300489.

Table[n! SeriesCoefficient[Log[1 + x]/(1 - x)^n, {x, 0, n}], {n, 0, 19}]
Table[n! Sum[(-1)^(k + 1) Binomial[2 n - k - 1, n - k]/k, {k, 1,  n}], {n, 0, 19}]
Join[{0}, Table[n^2 (2 (n - 1))! HypergeometricPFQ[{1, 1, 1 - n}, {2, 2 - 2 n}, -1]/n!, {n, 19}]]

a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(2*n-k-1,n-k)/k.

a(n) ~ log(3/2) * 2^(2*n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, May 05 2018

A336250 a(n) = (n!)^n * Sum_{k=1..n} (-1)^(k+1) / k^n.

Original entry on oeis.org

0, 1, 3, 197, 313840, 24191662624, 137300308036448256, 81994640912971156525105152, 6958651785463110878359050928999366656, 108902755985567407887534498777329973193771818418176, 395560567918154447056086270973712023435510589158871531520000000000

Offset: 0

Ilya Gutkovskiy, Jul 14 2020

nonn

Cf. A024167, A036740, A060943, A142999.

Table[(n!)^n Sum[(-1)^(k + 1)/k^n, {k, 1, n}], {n, 0, 10}]
Table[(n!)^n SeriesCoefficient[-PolyLog[n, -x]/(1 - x), {x, 0, n}], {n, 0, 10}]

a(n) = (n!)^n * sum(k=1, n, (-1)^(k+1) / k^n); \\ Michel Marcus, Jul 14 2020

a(n) = (n!)^n * [x^n] -polylog(n,-x) / (1 - x).

A347978 E.g.f.: 1/(1 + x)^(1/(1 - x)).

Original entry on oeis.org

1, -1, 0, -3, 4, -30, 186, -630, 11600, -26712, 1005480, -2581920, 117196872, -485308824, 17734457664, -131070696120, 3387342915840, -43890398953920, 801577841697216, -17363169328243392, 233460174245351040, -7968629225100337920, 84363134551361043840

Offset: 0

Ilya Gutkovskiy, Sep 22 2021

sign

Cf. A005727, A007120, A008405, A024167, A058312, A058313, A073478, A087761.

nmax = 22; CoefficientList[Series[1/(1 + x)^(1/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
A024167[n_] := n! Sum[(-1)^(k + 1)/k, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] A024167[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(1/(1+x)^(1/(1-x)))) \\ Michel Marcus, Sep 22 2021

E.g.f.: exp( Sum_{k>=1} x^k * Sum_{j=1..k} (-1)^j / j ).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * A024167(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * A073478(k) * a(n-k).

cp's OEIS Frontend

A078921 Signed variant of A077012.

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A087301 a(n) = n!*Sum_{i=1..n-1} (-1)^(i+1)/i.

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A346943 a(n) = a(n-1) + n*(n+1)*a(n-2) with a(0)=1, a(1)=1.

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A371685 Triangle read by rows: T(n, k) = n! * Sum_{j=0..n-1} binomial(k - 1, j) / (j + 1).

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A302582 a(n) = n! * [x^n] log(1 + x)/(1 - x)^n.

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A336250 a(n) = (n!)^n * Sum_{k=1..n} (-1)^(k+1) / k^n.

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A347978 E.g.f.: 1/(1 + x)^(1/(1 - x)).

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A346943 a(n) = a(n-1) + n(n+1)a(n-2) with a(0)=1, a(1)=1.