A000246 -id:A000246 - cp's OEIS frontend

A321268 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly two descents.

0, 0, 0, 0, 22, 172, 856, 3488, 12746, 43628, 143244, 457536, 1434318, 4438540, 13611136, 41473216, 125797010, 380341580, 1147318004, 3455325600, 10394291094, 31242645420, 93853769320, 281825553760, 846030314842, 2539248578732, 7620161662556, 22865518160768

Offset: 1

Sam Spiro, Nov 01 2018

Also the number of permutations of [n] of odd order whose M statistic (as defined in the Spiro paper) is equal to two.

			Some permutations counted by a(5) include 14253 and 34521.

Sam Spiro, Table of n, a(n) for n = 1..100
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (11,-50,122,-173,143,-64,12).

Cf. A000246, A005803, A321269, A303285, A177042.

Column k=2 of A321280.

a[1] = 0; a[n_] := 2n^2 - 2n - 1 - n 2^(n-1) - 2 Binomial[n, 3] + Sum[ Binomial[n, k] (2^k - 2k), {k, 0, n}];
Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Nov 11 2018 *)

a(n)={if(n<2, 0, 2*n^2 - 2*n - 1 - n*2^(n-1) - 2*binomial(n,3) + sum(k=0, n, binomial(n, k)*(2^k - 2*k)))} \\ Andrew Howroyd, Nov 01 2018

concat([0,0,0,0], Vec(2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)) + O(x^40))) \\ Colin Barker, Mar 07 2019

a(n) = 3*A008292(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.
a(n) =   A065826(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.
a(n) = 3^n-3*n*2^(n-1)-2*binomial(n,3)+4*binomial(n,2)-1 for n > 1.
From Colin Barker, Mar 07 2019: (Start)
G.f.: 2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)).
a(n) = 11*a(n-1) - 50*a(n-2) + 122*a(n-3) - 173*a(n-4) + 143*a(n-5) - 64*a(n-6) + 12*a(n-7) for n>8.
a(n) = -1 + 3^n - (16+9*2^n)*n/6 + 3*n^2 - n^3/3 for n>1.
(End)

A321269 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly three descents.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 604, 7296, 54746, 330068, 1756878, 8641800, 40298572, 180969752, 790697160, 3385019968, 14270283414, 59457742524, 245507935018, 1006678811272, 4105447763032, 16672235476128, 67482738851220, 272439143364672, 1097660274098482, 4415486996246052

Offset: 1

Sam Spiro, Nov 01 2018

Also the number of permutations of [n] of odd order whose M statistic (as defined in the Spiro paper) is equal to three.

			The permutations counted by a(7) include 1237654 and 17265243.

Alois P. Heinz, Table of n, a(n) for n = 1..1660
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (24,-260,1684,-7278,22172,-49004,79596,-95065,82508,-50616,20800,-5136,576).

Cf. A000246, A005803, A321268, A303285, A177042.

Column k=3 of A321280.

t[n_, k_] := Sum[(-1)^j (k - j)^n Binomial[n + 1, j], {j, 0, k}];
a[n_] := If[n<7, 0, 4 t[n-1, 4] - (Binomial[n, 3] - Binomial[n, 2] + 4) * 2^(n-2) - 22 Binomial[n, 5] + 16 Binomial[n, 4] - 4 Binomial[n, 3] + 2n];
Array[a, 30] (* Jean-François Alcover, Feb 29 2020, from Sam Spiro's 1st formula *)

concat([0,0,0,0,0,0], Vec(2*x^7*(302 - 3600*x + 18341*x^2 - 52006*x^3 + 89327*x^4 - 94728*x^5 + 61016*x^6 - 23368*x^7 + 5424*x^8 - 576*x^9) / ((1 - x)^6*(1 - 2*x)^4*(1 - 3*x)^2*(1 - 4*x)) + O(x^30))) \\ Colin Barker, Mar 07 2019

From Sam Spiro, Mar 07 2019: (Start)
a(n) = 4*A008292(n-1,4)-(binomial(n,3)-binomial(n,2)+4)*2^(n-2)-22*binomial(n,5)+16*binomial(n,4)-4*binomial(n,3)+2n for n>3.
a(n) = A065826(n-1,4)-(binomial(n,3)-binomial(n,2)+4)*2^(n-2)-22*binomial(n,5)+16*binomial(n,4)-4*binomial(n,3)+2n for n>3.
a(n) = 4^n-4*n*3^(n-1)+9*binomial(n,2)*2^(n-2)-binomial(n,3)*2^(n-2)-2^n-8*binomial(n,3)-22*binomial(n,5)+16*binomial(n,4)+2*n for n>3.
(End)
From Colin Barker, Mar 07 2019: (Start)
G.f.: 2*x^7*(302 - 3600*x + 18341*x^2 - 52006*x^3 + 89327*x^4 - 94728*x^5 + 61016*x^6 - 23368*x^7 + 5424*x^8 - 576*x^9) / ((1 - x)^6*(1 - 2*x)^4*(1 - 3*x)^2*(1 - 4*x)).
a(n) = 24*a(n-1) - 260*a(n-2) + 1684*a(n-3) - 7278*a(n-4) + 22172*a(n-5) - 49004*a(n-6) + 79596*a(n-7) - 95065*a(n-8) + 82508*a(n-9) - 50616*a(n-10) + 20800*a(n-11) - 5136*a(n-12) + 576*a(n-13) for n>16.
(End)

More terms from Alois P. Heinz, Nov 01 2018

A331618 E.g.f.: exp(1 / (1 - arctanh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 15, 97, 785, 7523, 83615, 1053281, 14838177, 230832867, 3929944623, 72633052545, 1447981700529, 30960823851267, 706676217730239, 17145815895371073, 440594781536265537, 11952178787661839427, 341291300477569866831, 10231558345117929439521

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Cf. A000246, A080833, A202139, A296676, A331610, A331615, A331616, A331617.

nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcTanh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A296676[0] = 1; A296676[n_] := A296676[n] = Sum[Binomial[n, k] If[OddQ[k], (k - 1)!, 0] A296676[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A296676[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

seq(n)={Vec(serlaplace(exp(1/(1 - atanh(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A296676(k) * a(n-k).

a(n) ~ (exp(2) + 1)^(n - 1/4) * n^(n - 1/4) / ((exp(2) - 1)^(n + 1/4) * exp(n - 4*exp(1)*sqrt(n/(exp(4) - 1)) - 2/(exp(4) - 1) - 1/2)). - Vaclav Kotesovec, Jan 26 2020

A086325 Let u(1)=0, u(2)=1, u(k)=u(k-1)+u(k-2)/(k-2); then a(n)=n!*u(n).

Original entry on oeis.org

0, 2, 6, 36, 220, 1590, 12978, 118664, 1201464, 13349610, 161530270, 2114578092, 29780308116, 448995414686, 7215997736010, 123153028027920, 2224451568754288, 42395429898611154, 850263899633257014, 17900292623858042420, 394701452356069835340, 9096928711444657157382, 218739785834282892557026

Offset: 1

N. J. A. Sloane. This sequence appeared in the 1973 "Handbook", but was then omitted from the database. Resubmitted by Benoit Cloitre, Aug 30 2003. Entry revised by N. J. A. Sloane, Jun 11 2012

nonn

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263, Table 7.5.1, row 3.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210, Table 3, Three-line Latin rectangles.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

Cf. A000246, A000274, A000166.

a:=n->n!*add((-1)^k/k!, k=0..n): seq(a(n)*n, n=1..19); # Zerinvary Lajos, Dec 18 2007
with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*fibonacci(2,n), n=1..19); # Zerinvary Lajos, Jun 11 2008
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*n, n=1..19); # Zerinvary Lajos, Jun 11 2008

Table[Subfactorial[n]*n, {n, 1, 19}] (* Zerinvary Lajos, Jul 09 2009 *)

a(n) = n*((n! + 1)\exp(1)); \\ Indranil Ghosh, Apr 13 2017

a(n) = ceiling(n*n!/e) - (1-(-1)^n)/2.
E.g.f.: x^2*exp(-x)/(1-x)^2. - Vladeta Jovovic, Nov 20 2003
a(n) = n*floor((n!+1)/e). [Gary Detlefs, Jul 13 2010]
a(n) = n * A000166(n). [Joerg Arndt, Jul 09 2012]
G.f.: x*f'(x), where f(x) = 1/(1 + x) + Sum_{k>=1} k^k*x^k/(1 + (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017

A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Johannes W. Meijer, Jul 27 2016

nonn

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.
To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the inverse multinomial transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.
The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)

Richard P. Feynman, QED, The strange theory of light and matter, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.
Wikipedia, Feynman diagram

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.

A290384 Number of ordered set partitions of [n] such that the smallest element of each block is odd.

Original entry on oeis.org

1, 1, 1, 3, 5, 23, 57, 355, 1165, 9135, 37313, 352667, 1723605, 19063207, 108468169, 1374019539, 8920711325, 127336119839, 928899673425, 14751357906571, 119445766884325, 2088674728868631, 18588486479073881, 354892573941671363, 3443175067395538605

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

All terms are odd.

			a(3) = 3: 123, 12|3, 3|12.
a(4) = 5: 1234, 124|3, 3|124, 12|34, 34|12.

Alois P. Heinz, Table of n, a(n) for n = 0..494
Wikipedia, Partition of a set

Cf. A000246, A000670, A019538, A290383.

b:= proc(n, m, t) option remember; `if`(n=0, m!,
      add(b(n-1, max(m, j), 1-t), j=1..m+1-t))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);

b[n_, m_, t_]:=b[n, m, t]=If[n==0, m!, Sum[b[n - 1, Max[m, j], 1 - t], {j, m + 1 - t}]]; Table[b[n, 0, 0], {n, 0, 50}] (* Indranil Ghosh, Jul 30 2017 *)

{ A290384(n) = (n==0) + sum(m=0,n, sum(k=1,m+1, stirling(m,k-1,2)*(k-1)! * stirling(n-m,k,2)*k! * (-1)^(m+k+1))); } \\ Max Alekseyev, Sep 28 2021

{ A290384(n) = polcoef(1 + sum(k=1,n, (-1)^(k-1) / binomial(-1/x-1,k-1) / binomial(1/x-1,k) + O(x^(n+1)) ), n); } \\ Max Alekseyev, Sep 23 2021

For n>=1, a(n) = Sum_{m=0..n} Sum_{k=1..m+1} (-1)^(m+k+1) * S(m,k-1) * (k-1)! * S(n-m,k) * k! = Sum_{m=0..n} Sum_{k=1..m+1} (-1)^(m+k+1) * A019538(m,k-1) * A019538(n-m,k). - Max Alekseyev, Sep 28 2021

G.f.: 1 + Sum_{k >= 1} (-1)^(k-1) / binomial(-1/x-1,k-1) / binomial(1/x-1,k). - Max Alekseyev, Sep 23 2021

A293193 a(n) = n! * [x^n] exp(n*arctanh(x)).

Original entry on oeis.org

1, 1, 4, 33, 384, 5745, 105120, 2273985, 56770560, 1606455585, 50810457600, 1776342038625, 68018227200000, 2831056111508625, 127263741276672000, 6144722994699914625, 317153340121153536000, 17425813179911850338625, 1015487574490836615168000, 62559446107077837491198625

Offset: 0

Ilya Gutkovskiy, Oct 02 2017

nonn

Cf. A000246.

Table[n! SeriesCoefficient[Exp[n ArcTanh[x]], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[n Sum[x^(2 k + 1)/(2 k + 1), {k, 0, Infinity}]], {x, 0, n}], {n, 0, 19}]

a(n) ~ phi^(5*n/2) * n^n / (5^(1/4) * exp(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 02 2017

A296789 Expansion of e.g.f. exp(x*arctanh(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 504, 24464, 1959840, 234852672, 39370660224, 8799246209280, 2528787321598464, 908585701684024320, 399070678264750356480, 210373049449102957645824, 131083661069772517440921600, 95304505860052894815543705600, 79961055068441273887848131297280

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

nonn

			exp(x*arctanh(x)) = 1 + 2*x^2/2! + 20*x^4/4! + 504*x^6/6! + 24464*x^8/8! + ...

Cf. A000246, A009252, A009273, A010050, A166356, A259647, A293193, A296787, A296788.

nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTanh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Exp[x (Log[1 + x] - Log[1 - x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] exp(x*arctanh(x)).

a(n) ~ 2^(2*n + 2) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Dec 21 2017

A304001 Number of permutations of [n] whose up-down signature has a nonnegative total sum.

Original entry on oeis.org

1, 1, 1, 5, 12, 93, 360, 3728, 20160, 259535, 1814400, 27820524, 239500800, 4251096402, 43589145600, 877606592736, 10461394944000, 235288904377275, 3201186852864000, 79476406782222500, 1216451004088320000, 33020655481590446318, 562000363888803840000

Offset: 0

Alois P. Heinz, May 04 2018

nonn

The up-down signature has (+1) for each ascent and (-1) for each descent.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Bisections give: A002674 (even part), A179457(2n+1,n+1) (odd part).

Cf. A000246 (for nonnegative partial sums), A006551 (total sums are 0 or 1), A008292, A303287.

b:= proc(u, o, t) option remember; (n->
     `if`(t>=n, n!, `if`(t<-n, 0,
      add(b(u-j, o+j-1, t-1), j=1..u)+
      add(b(u+j-1, o-j, t+1), j=1..o))))(u+o)
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 0), j=1..n)):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> `if`(irem(n, 2, 'r')=0, ceil(n!/2),
         add(combinat[eulerian1](n, j), j=0..r)):
seq(a(n), n=0..25);

Eulerian1[n_, k_] := If[k == 0, 1, If[n == 0, 0, Sum[(-1)^j (k - j + 1)^n Binomial[n + 1, j], {j, 0, k + 1}]]];
a[n_] := Module[{r, m}, {r, m} = QuotientRemainder[n, 2]; If[m == 0, Ceiling[n!/2], Sum[Eulerian1[n, j], {j, 0, r}]]];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 26 2021, after 2nd Maple program *)

A317618 Expansion of e.g.f. sqrt((1 - x)/(1 - 3*x)).

Original entry on oeis.org

1, 1, 5, 39, 417, 5685, 94365, 1847475, 41686785, 1065288105, 30411314325, 959236098975, 33129890726625, 1243507150410525, 50401090111697325, 2193907232242600875, 102075654396429338625, 5055304328553234380625, 265522264682686831945125, 14742355948224269570580375

Offset: 0

Ilya Gutkovskiy, Aug 01 2018

nonn

Lah transform of A001147.

Robert Israel, Table of n, a(n) for n = 0..380
N. J. A. Sloane, Transforms

Cf. A000246, A001147, A002866, A052563, A084262.

a:=series(sqrt((1 - x)/(1 - 3*x)), x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # Paolo P. Lava, Mar 26 2019

nmax = 19; CoefficientList[Series[Sqrt[(1 - x)/(1 - 3*x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] (2 k - 1)!! n!/k!, {k, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! Hypergeometric2F1[3/2, 1 - n, 2, -2], {n, 19}]]

my(x='x + O('x^25)); Vec(serlaplace(sqrt((1 - x)/(1 - 3*x)))) \\ Michel Marcus, Mar 26 2019

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*(2*k-1)!!*n!/k!.
a(n) ~ 2 * 3^(n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 26 2019
D-finite with recurrence: (3*n^2 + 3*n)*a(n) + (-5 - 4*n)*a(n + 1) + a(n + 2)=0. - Robert Israel, Mar 26 2019

cp's OEIS Frontend

A321268 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly two descents.

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A321269 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly three descents.

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A331618 E.g.f.: exp(1 / (1 - arctanh(x)) - 1).

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A086325 Let u(1)=0, u(2)=1, u(k)=u(k-1)+u(k-2)/(k-2); then a(n)=n!*u(n).

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A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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A290384 Number of ordered set partitions of [n] such that the smallest element of each block is odd.

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A293193 a(n) = n! * [x^n] exp(n*arctanh(x)).

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