A003724 -id:A003724 - cp's OEIS frontend

A307978 Expansion of e.g.f. exp((sinh(x) - sin(x))/2).

1, 0, 0, 1, 0, 0, 10, 1, 0, 280, 120, 1, 15400, 17160, 2080, 1401401, 3203200, 1290640, 190623040, 775975201, 712150400, 36321556720, 239000886400, 413465452401, 9339501072000, 91625659447400, 266045692290560, 3216459513124001, 42923384190336000, 193108117771690680

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Number of partitions of n-set into blocks congruent to 3 mod 4.

Seiichi Manyama, Table of n, a(n) for n = 0..608

Cf. A002017, A003724, A004767, A005046, A013369, A035462, A291975, A306347, A307979.

nmax = 29; CoefficientList[Series[Exp[(Sinh[x] - Sin[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{3}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 29}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sinh(x)-sin(x))/2))) \\ Seiichi Manyama, Mar 17 2022

a(n) = if(n==0, 1, sum(k=0, (n-3)\4, binomial(n-1, 4*k+2)*a(n-4*k-3))); \\ Seiichi Manyama, Mar 17 2022

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/4)} binomial(n-1,4*k+2) * a(n-4*k-3). - Seiichi Manyama, Mar 17 2022

A009227 Expansion of e.g.f.: exp(sinh(x))/exp(x).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 10, 1, 56, 281, 246, 4621, 16412, 53197, 564642, 1937937, 13309648, 100397649, 454215214, 4082253589, 26498068420, 174791970677, 1575851086778, 10628056916313, 91523101970104, 788580099169337, 6237722665351750, 60190551618214941

Offset: 0

R. H. Hardin

Number of partitions of n-set in which block sizes are odd and greater than 1. - Vladeta Jovovic, Aug 23 2007

Cf. A003724, A009209.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, 2*j)*a(n-2*j-1), j=1..(n-1)/2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 09 2022

With[{nn=30},CoefficientList[Series[Exp[Sinh[x]]/Exp[x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 18 2015 *)

my(x='x+O('x^30)); Vec(serlaplace(exp(sinh(x))/exp(x))) \\ Michel Marcus, Apr 09 2022

a(0) = 1; a(n) = Sum_{k=1..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Apr 09 2022

Extended and signs tested by Olivier Gérard, Mar 15 1997

Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Jan 18 2015

A011800 Number of labeled forests of n nodes each component of which is a path.

Original entry on oeis.org

1, 1, 2, 7, 34, 206, 1486, 12412, 117692, 1248004, 14625856, 187638716, 2614602112, 39310384192, 634148436104, 10923398137576, 200069534481616, 3882002527006352, 79535575126745632, 1715658099715217584

Offset: 0

Herbert S. Wilf

Denote the bivariate exponential g.f. by g(x,y)=exp(y*f(x)) where f(x)=(2x-x^2)/(2-2x). Then this sequence is the row sums of the array defined by the g.f. The differential dg/dy = f(x)*exp(y*f(x)) is the exponential generating function for an array with row sums in A201720. - R. J. Mathar, Jun 27 2022

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (3.3.6).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(d).

T. D. Noe, Table of n, a(n) for n = 0..100
Tobias Boege, Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.
J. Rasku, T. Karkkainen, P. Hotokka, Solution Space Visualization as a Tool for Vehicle Routing Algorithm Development, Proc. FORS-40, pp. 9-12, 2013.

Function[ esl, esl*Array[ Factorial, Length[ esl ], 0 ] ][ CoefficientList[ Series[ Exp[ x+x^2/(2-2x) ], {x, 0, 20} ], x ] ] (* Olivier Gérard *)
With[{nn=20},CoefficientList[Series[Exp[x+x^2/(2*(1-x))],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 13 2019 *)

a(n):=n!*sum(sum(binomial(k,n-k-i)*binomial(k+i-1,k-1)*2^(-n+k+i)*(-1)^(n-k-i),i,0,n-k)/(k!),k,1,n); /* Vladimir Kruchinin, Nov 25 2012 */

E.g.f.: exp[ x + x^2/(2*(1 - x)) ].
a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A003724(k). - Vladeta Jovovic, Oct 19 2003
Recurrence: 2*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-1)^2*a(n-2) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 07 2012
a(n) ~ 2^(-3/4)*exp(sqrt(2*n)-n+1/4)*n^(n-1/4). - Vaclav Kotesovec, Oct 07 2012
a(n) = n!*Sum_{k=1..n} (Sum_{i=0..n-k} binomial(k,n-k-i)*binomial(k+i-1,k-1)*2^(-n+k+i)*(-1)^(n-k-i))/k!, n > 0, a(0) = 1. - Vladimir Kruchinin, Nov 25 2012

A115276 Number of partitions of {1,...,n} into block sizes not a multiple of 4.

Original entry on oeis.org

1, 1, 2, 5, 14, 47, 173, 702, 3124, 14901, 76405, 417210, 2411466, 14731095, 94573911, 636575050, 4480990936, 32887804361, 251236573561, 1993395483746, 16397468177406, 139634290253907, 1229013163330947, 11166172488138322, 104593176077399652

Offset: 0

Christian G. Bower, Jan 18 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A001935, A003724, A115275, A115277.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      irem(j, 4)=0, 0, binomial(n-1, j-1)*a(n-j)), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2015

a[n_] := a[n] = If[n == 0, 1, Sum[If[Mod[j, 4] == 0, 0, Binomial[n - 1, j - 1]*a[n - j]], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)

E.g.f.: exp(sinh(x)+(cosh(x)-cos(x))/2).

A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.

Original entry on oeis.org

1, 1, 3, 17, 137, 1457, 19355, 308961, 5766353, 123285153, 2972114803, 79782059249, 2360417058521, 76319622510289, 2677629295171979, 101318751122847297, 4113158120834726049, 178328823993199602241, 8223999403291995520995, 401989145900847087408849

Offset: 0

Paul D. Hanna, Nov 20 2012

nonn

Compare to the LambertW identity: Sum_{n>=0} (n+1)^(n-1)*exp(-n*x)*x^n/n! = exp(x).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 137*x^4/4! + 1457*x^5/5! +...
where
A(x) = 1 + sinh(x) + 3^1*sinh(x)^2/2! + 4^2*sinh(x)^3/3! + 5^3*sinh(x)^4/4! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A003724, A136630, A162650.

Cf. A238085.

CoefficientList[Series[-LambertW[-Sinh[x]]/Sinh[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)

{a(n)=n!*polcoeff(sum(k=0,n,(k+1)^(k-1)*sinh(x + x*O(x^n))^k/k!),n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: LambertW(-sinh(x)) / (-sinh(x)).
a(n) ~ (1+exp(2))^(1/4) * n^(n-1) / (exp(n-1) * log(exp(-1) +sqrt(1+exp(-2)))^(n-1/2)). - Vaclav Kotesovec, Jul 08 2013
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A136630(n,k). - Seiichi Manyama, Feb 15 2025

A275679 Number of set partitions of [n] with alternating block size parities.

Original entry on oeis.org

1, 1, 1, 4, 3, 20, 43, 136, 711, 1606, 12653, 36852, 250673, 1212498, 6016715, 45081688, 196537387, 1789229594, 8963510621, 76863454428, 512264745473, 3744799424978, 32870550965259, 219159966518160, 2257073412153459, 15778075163815474, 165231652982941085

Offset: 0

Alois P. Heinz, Aug 05 2016

nonn

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 3: 1234, 1|23|4, 1|24|3.
a(5) = 20: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|2|45, 145|23, 14|235, 15|234, 1|2345, 14|2|35, 15|2|34.

Alois P. Heinz, Table of n, a(n) for n = 0..592

Cf. A003724, A005046, A007837, A038041, A275309, A275310, A275311, A275312, A275313, A286076, A361804.

b:= proc(n, t) option remember; `if`(n=0, 1, add(
      `if`((i+t)::odd, b(n-i, 1-t)*binomial(n-1, i-1), 0), i=1..n))
    end:
a:= n-> `if`(n=0, 1, b(n, 0)+b(n, 1)):
seq(a(n), n=0..35);

b[n_, t_] := b[n, t] = If[n==0, 1, Sum[If[OddQ[i+t], b[n-i, 1-t] * Binomial[n-1, i-1], 0], {i, 1, n}]]; a[n_] := If[n==0, 1, b[n, 0] + b[n, 1]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

A331608 E.g.f.: exp(1 / (1 - sinh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 14, 85, 632, 5559, 56352, 645929, 8252352, 116189291, 1786361216, 29764770941, 534082233856, 10264484355103, 210312181051392, 4575364233983057, 105310034714202112, 2556360647841415379, 65261358332774277120, 1747713179543456515749

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Cf. A003704, A003724, A006154, A075729, A331607, A331611.

nmax = 20; CoefficientList[Series[Exp[1/(1 - Sinh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A006154[n_] := Sum[Sum[(-1)^j (k - 2 j)^n Binomial[k, j]/2^k, {j, 0, k}], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A006154[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A006154(k) * a(n-k).

a(n) ~ exp(1/(2^(3/2) * log(1 + sqrt(2))) - 3/4 + 2^(3/4) * sqrt(n) / sqrt(log(1 + sqrt(2))) - n) * n^(n - 1/4) / (2^(5/8) * log(1 + sqrt(2))^(n + 1/4)). - Vaclav Kotesovec, Jan 27 2020

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

Original entry on oeis.org

1, 1, 3, 7, 49, 321, 2131, 19783, 195777, 2101249, 25721731, 340358151, 4902173233, 75688032577, 1253701725459, 22347046050631, 418439924732161, 8318748086461953, 175769214730290307, 3871849719998940679, 89734800330818444721, 2187944831367633226561

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A000726, A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := x*(1+x)/(1-x^3) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(0=
      irem(j, 3), 0, a(n-j)*j!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

CoefficientList[Series[E^(x*(1+x)/(1-x^3)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: exp(x*(1+x)/(1-x^3)).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + 2*(n-3)*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 6^(-1/4) * n^(n-1/4) * exp(2/3*sqrt(6*n)-n) * (1 - 43/(48*sqrt(6*n))). - Vaclav Kotesovec, Sep 25 2013

2 more terms from R. J. Mathar, Feb 06 2008

A333882 Expansion of e.g.f. exp(Sum_{k>=0} x^(5k + 1) / (5k + 1)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 1399, 7801, 45410, 216581, 853218, 2896002, 11708734, 79817500, 615700986, 4012571831, 21538473686, 98707812691, 501634082800, 3983368886226, 37404203343457, 305886831698593, 2069143637726674, 11924094649669375

Offset: 0

Ilya Gutkovskiy, Apr 08 2020

nonn

Number of partitions of n-set into blocks congruent to 1 mod 5.

Cf. A000110, A003724, A306347, A333881, A333883.

nmax = 29; CoefficientList[Series[Exp[Sum[x^(5 k + 1)/(5 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 5]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 29}]
nmax = 30; CoefficientList[Series[Exp[x*HypergeometricPFQ[{}, {2/5, 3/5, 4/5, 6/5}, x^5/3125]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} binomial(n-1,5*k) * a(n-5*k-1). - Seiichi Manyama, Sep 22 2023

A351937 Expansion of e.g.f. exp( (sinh(x) + x*cosh(x)) / 2 ).

Original entry on oeis.org

1, 1, 1, 3, 9, 24, 99, 418, 1769, 9320, 49541, 278912, 1764825, 11319784, 77850287, 570610472, 4290387409, 34316005632, 285335249065, 2455224885440, 22165590003849, 206191758121856, 1989511661589435, 19903718061574144, 204795484665487865, 2179948112062667392

Offset: 0

Ilya Gutkovskiy, Feb 26 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..568

Cf. A000248, A003724, A003727, A349103, A349105.

nmax = 25; CoefficientList[Series[Exp[(Sinh[x] + x Cosh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] (k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

my(x='x+O('x^30)); Vec(serlaplace(exp((sinh(x) + x*cosh(x))/2))) \\ Michel Marcus, Feb 26 2022

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * (k+1) * a(n-2*k-1).

cp's OEIS Frontend

A307978 Expansion of e.g.f. exp((sinh(x) - sin(x))/2).

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PARI

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A009227 Expansion of e.g.f.: exp(sinh(x))/exp(x).

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Extensions

A011800 Number of labeled forests of n nodes each component of which is a path.

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Maxima

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A115276 Number of partitions of {1,...,n} into block sizes not a multiple of 4.

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A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.

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Mathematica

PARI

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A275679 Number of set partitions of [n] with alternating block size parities.

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Maple

Mathematica

A331608 E.g.f.: exp(1 / (1 - sinh(x)) - 1).

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Mathematica

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A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

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A333882 Expansion of e.g.f. exp(Sum_{k>=0} x^(5k + 1) / (5k + 1)!).