A052895 -id:A052895 - cp's OEIS frontend

A006531 Semiorders on n elements.

1, 1, 3, 19, 183, 2371, 38703, 763099, 17648823, 468603091, 14050842303, 469643495179, 17315795469063, 698171064855811, 30561156525545103, 1443380517590979259, 73161586346500098903, 3961555049961803092531, 228225249142441259147103, 13938493569348563803135339

Offset: 0

N. J. A. Sloane

Labeled semiorders on n elements: (1+3) and (2+2)-free posets. - Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002

Labeled incomplete binary trees (every vertex has a left child, a right child, neither, or both) in which every vertex with a right child but no left child has a label greater than the label of its right child. - Ira M. Gessel, Nov 09 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.30.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. L. Chandon, J. LeMaire and J. Pouget, Dénombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
J. L. Chandon, J. LeMaire and J. Pouget, Enumeration of semiorders on a finite set, Preprint (English) of "Dénombrement des quasi-ordres sur un ensemble fini".
J. L. Chandon, Letter to N. J. A. Sloane, May 1981
Julie Christophe, Jean-Paul Doignon and Samuel Fiorini, Counting Biorders, J. Integer Seqs., Vol. 6, 2003.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
Yan X. Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

Cf. A000108 (unlabeled semiorders: Catalan numbers), A052895.

A006531 := n->add(stirling2(n,k)*k!*A001006(k-1),k=1..n);

m[n_] := m[n] = m[n-1] + Sum[ m[k]*m[n-k-2], {k, 0, n-2}]; m[0] = a[0] = 1; a[n_] := Sum[ StirlingS2[n, k]*k!*m[k-1], {k, 1, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 24 2012, after Maple *)

{a(n)=polcoeff(sum(m=0, n, (2*m)!/(m+1)!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */

E.g.f.: C(1-exp(-x)), where C(x) = (1 - sqrt(1 - 4*x)) / (2*x) is the ordinary g.f. for the Catalan numbers A000108. [corrected by Joel B. Lewis, Mar 29 2011]
a(n) = Sum_{k=1..n} S(n, k) * k! * M(k-1), S(n, k): Stirling number of the second kind, M(n): Motzkin number, A001006. - Detlef Pauly, Jun 06 2002
O.g.f.: Sum_{n>=1} (2*n)!/(n+1)! * x^n / Product_{k=0..n} (1+k*x). - Paul D. Hanna, Jul 20 2011
a(n) ~ n! * sqrt(3)*(log(4/3))^(1/2-n)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
E.g.f.: 1/(1 + (exp(-x) - 1)/(1 + (exp(-x) - 1)/(1 + (exp(-x) - 1)/(1 + (exp(-x) - 1)/(1 + ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 18 2017
From Peter Bala, Jan 15 2018: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*Catalan(k)*k!*Stirling2(n,k). Cf. A052895.
Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is Euler's totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 3, 9, 3, 1, 3, 9, 3, ...) with an apparent period 1, 3, 9, 3 of length 4 = phi(10) beginning at a(1). (End)
Consider the transformation of a sequence u given by T(u)(m) = (-1)^m*Sum_{n=0..m} (u(n)/(n+1))*(Sum_{k=0..n}(-1)^k*binomial(n,k)*k^m). If u(n) = 1 then T(u)(n) = Bernoulli(n) (with Bernoulli(1) = 1/2), if u(n) = binomial(2*n,n) then T(u)(n) = a(n). - Peter Luschny, Jul 09 2020

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(x))).

Original entry on oeis.org

1, 1, 6, 57, 796, 14785, 344046, 9640225, 316255416, 11896233345, 504918768250, 23874754106401, 1244712973780068, 70940791877082049, 4388291507415513894, 292823509879910802465, 20966854494419642792176, 1603540841320336494905089, 130464295561360336835272050

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

nonn

Inverse binomial transform of A194471.

Seiichi Manyama, Table of n, a(n) for n = 0..356

Cf. A000108, A006531, A052895, A194471, A213644, A295239.

a:=series(2/(1+sqrt(1-4*x*exp(x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # Paolo P. Lava, Mar 27 2019

nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 18}]

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Aug 15 2023

E.g.f.: 1/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2*(1 + LambertW(1/4))) * n^(n-1) / ((LambertW(1/4))^n * exp(n)). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+1,k)/( (2*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Aug 15 2023

A367161 E.g.f. satisfies A(x) = 1 + A(x)^3 * (exp(x) - 1).

Original entry on oeis.org

1, 1, 7, 91, 1795, 47851, 1612027, 65731891, 3148530595, 173319612571, 10782796483147, 748237171338691, 57299882326956595, 4800323120225595691, 436719009263680421467, 42878536726317406241491, 4519124182661042439577795, 508885588456024192452993211

Offset: 0

Seiichi Manyama, Nov 07 2023

nonn

Cf. A367155, A367158, A367164.

Cf. A052895.

Table[Sum[(3*k)!/(2*k+1)! * StirlingS2[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 10 2023 *)

a(n) = sum(k=0, n, (3*k)!/(2*k+1)!*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (3*k)!/(2*k+1)! * Stirling2(n,k).

a(n) ~ sqrt(93) * n^(n-1) / (2^(5/2) * log(31/27)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Nov 10 2023

A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

Original entry on oeis.org

1, 1, 8, 64, 824, 12968, 252720, 5789712, 153169440, 4589004192, 153643615872, 5684390364288, 230307823878144, 10141452865049088, 482259966649655808, 24630247225278881280, 1344614199041549399040, 78137673004382654223360

Offset: 1

Vladeta Jovovic, Oct 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..372
Index entries for reversions of series

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287, A293604.

Cf. A370462, A370463.

Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)

x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ G. C. Greubel, May 24 2017

a(n) = Sum_{k=1..n} Stirling1(n, k)*k!*Catalan(k-1).
a(n) ~ n! / (2*exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (log(1 + x)) / (1 - A(x)).
E.g.f.: Series_Reversion( exp(x * (1 - x)) - 1 ). (End)

A377454 E.g.f. satisfies A(x) = 1/(1 - A(x) * (exp(x) - 1))^4.

Original entry on oeis.org

1, 4, 56, 1384, 50216, 2422024, 146279816, 10633540264, 904699882856, 88234503004744, 9707888368200776, 1189726637663987944, 160741241332049376296, 23738834426406792534664, 3804763374380021378204936, 657774175587674349626736424, 122016250347540672925706274536

Offset: 0

Seiichi Manyama, Oct 28 2024

nonn

Cf. A052895, A377452, A377453.

Cf. A377451.

a(n) = 4*sum(k=0, n, (5*k+3)!/(4*k+4)!*stirling(n, k, 2));

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377451.

a(n) = 4 * Sum_{k=0..n} (5*k+3)!/(4*k+4)! * Stirling2(n,k).

A087152 Expansion of (1-sqrt(1-4*log(1+x)))/log(1+x)/2.

Original entry on oeis.org

1, 3, 20, 194, 2554, 42226, 843744, 19769256, 531768120, 16152296424, 546895099200, 20425461026736, 834215500905552, 36988602430554576, 1769524998544143360, 90851799797294235264, 4982968503277896871296

Offset: 1

Vladeta Jovovic, Oct 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..370

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287.

Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/Log[1+x]/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)

x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/log(1+x)/2 - 1)) \\ G. C. Greubel, May 24 2017

a(n) = Sum_{k=0..n} Stirling1(n, k)*k!*Catalan(k).

a(n) ~ 2*n! / (exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015

A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4xexp(x))).

Original entry on oeis.org

1, -1, 2, -9, 68, -705, 9234, -146209, 2717000, -57986433, 1397949830, -37576332321, 1114326129564, -36141571087297, 1272713716466906, -48360394499269665, 1972269941821097744, -85929979225787811585, 3983422470176606823054, -195765982110500512129057

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

sign

Cf. A000108, A006531, A052895, A295238.

a:=series(2/(1+sqrt(1+4*x*exp(x))),x=0,20): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019

nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 + 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(-1)^m (m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 19}]

a(n) = n!*sum(k=0, n, (-1)^k*k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Oct 30 2024

E.g.f.: 1/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + ...))))), a continued fraction.
a(n) ~ sqrt(2*(1+LambertW(-1/4))) * n^(n-1) / (exp(n) * (LambertW(-1/4))^n). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} (-1)^k * k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Oct 30 2024

A377453 E.g.f. satisfies A(x) = 1/(1 - A(x) * (exp(x) - 1))^3.

Original entry on oeis.org

1, 3, 33, 639, 18177, 687663, 32585793, 1858893039, 124128928257, 9502575055983, 820716875385153, 78959656742857839, 8375163183606235137, 971063889443489669103, 122194096152956362997313, 16586054767142612489229039, 2415658529914018225123490817, 375779208568915395913102663023

Offset: 0

Seiichi Manyama, Oct 28 2024

nonn

Cf. A052895, A377452, A377454.

Cf. A377450.

a(n) = 3*sum(k=0, n, (4*k+2)!/(3*k+3)!*stirling(n, k, 2));

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A377450.

a(n) = 3 * Sum_{k=0..n} (4*k+2)!/(3*k+3)! * Stirling2(n,k).

A377452 E.g.f. satisfies A(x) = 1/(1 - A(x) * (exp(x) - 1))^2.

Original entry on oeis.org

1, 2, 16, 224, 4612, 126392, 4340836, 179534504, 8693925172, 482731239032, 30243460133956, 2110849596096584, 162438922745208532, 13665129603889106072, 1247684652874279407076, 122885960933254703151464, 12987106624622962667192692, 1466014441678589235669027512

Offset: 0

Seiichi Manyama, Oct 28 2024

nonn

Cf. A052895, A377453, A377454.

Cf. A367161.

a(n) = 2*sum(k=0, n, (3*k+1)!/(2*k+2)!*stirling(n, k, 2));

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367161.

a(n) = 2 * Sum_{k=0..n} (3*k+1)!/(2*k+2)! * Stirling2(n,k).

A377716 E.g.f. satisfies A(x) = (1 + (exp(x) - 1) * A(x))^2.

Original entry on oeis.org

1, 2, 12, 116, 1584, 28172, 619872, 16289996, 498428544, 17417438252, 684759380832, 29925135793676, 1439467532867904, 75591768584407532, 4303733247493423392, 264082643528395550156, 17375242687235713361664, 1220318925238762558532012, 91128522664443184593699552

Offset: 0

Seiichi Manyama, Nov 04 2024

Cf. A000670, A377717.

Cf. A052895, A377692.

a(n) = 2*sum(k=0, n, (2*k+1)!/(k+2)!*stirling(n, k, 2));

E.g.f.: 4/(1 + sqrt(5 - 4*exp(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052895.
a(n) = 2 * Sum_{k=0..n} (2*k+1)!/(k+2)! * Stirling2(n,k).
a(n) ~ 2^(5/2) * sqrt(5) * n^(n-1) / (exp(n) * log(5/4)^(n - 1/2)). - Vaclav Kotesovec, Aug 27 2025

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A006531 Semiorders on n elements.

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A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*exp(x))).

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A367161 E.g.f. satisfies A(x) = 1 + A(x)^3 * (exp(x) - 1).

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A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

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A377454 E.g.f. satisfies A(x) = 1/(1 - A(x) * (exp(x) - 1))^4.

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A087152 Expansion of (1-sqrt(1-4*log(1+x)))/log(1+x)/2.

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A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4*x*exp(x))).

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A377453 E.g.f. satisfies A(x) = 1/(1 - A(x) * (exp(x) - 1))^3.

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A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(x))).

A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4xexp(x))).