A135753 -id:A135753 - cp's OEIS frontend

A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

1, 1, 3, 13, 87, 841, 11643, 227893, 6285807, 243593041, 13262556723, 1014466283293, 109128015915207, 16521353903210521, 3524056001906654763, 1059868947134489801413, 449831067019305308555487, 269568708630308018001547681, 228228540531327778410439620963

Offset: 0

N. J. A. Sloane

Labeled posets where for all a,b,c in the set, do not have a

Number of labeled digraphs with n vertices with no directed path of length 2. Number of n X n {0,1} matrices A such that A^2 = 0. - Michael Somos, Jul 28 2013

Number of relations on n labeled nodes that are simultaneously transitive and antitransitive. - Peter Kagey, Feb 14 2021

			1 + x + 3*x^2 + 13*x^3 + 87*x^4 + 841*x^5 + 11643*x^6 + 227893*x^7 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint, Australasian Joint Conference on Artificial Intelligence AI 2015: Advances in Artificial Intelligence, pp. 411-423.
Zvi Rosen and Yan X. Zhang, Convex Neural Codes in Dimension 1, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
Index entries for sequences related to posets

Cf. A002031, A047863, A052332, A001833, A048194.
Row sums of A052296.
Cf. variants: A135753, A135754.

A001831 := proc(n)
    add(binomial(n,k)*(2^k-1)^(n-k),k=0..n) ;
end proc:
seq(A001831(n),n=0..10) ; # R. J. Mathar, Mar 08 2021

Join[{1}, Table[Sum[Binomial[n,k](2^k-1)^(n-k),{k,n}],{n,20}]] (* Harvey P. Dale, Jan 05 2012 *)

{a(n)=n!*polcoeff(sum(k=0,n,exp((2^k-1)*x)*x^k/k!),n)} \\ Paul D. Hanna, Nov 27 2007

{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k-1)*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Sep 15 2009

a(n) = Sum((-1)^k*C(n, k)*A047863(k), k=0..n).
a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-1)^(n-k). - Vladeta Jovovic, Apr 04 2003
E.g.f.: Sum_{n>=0} exp((2^n-1)*x) * x^n/n!. - Paul D. Hanna, Nov 27 2007 [correction made by Paul D. Hanna, Mar 08 2021]
O.g.f.: Sum_{n>=0} x^n/(1 - (2^n - 1)*x)^(n+1) = Sum_{n>=0} a(n)*x^n. - Paul D. Hanna, Sep 15 2009
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999
Last 4 terms corrected by Vladeta Jovovic, Apr 04 2003
Comments corrected by Joel B. Lewis, Mar 28 2011

A135754 E.g.f.: A(x) = Sum_{n>=0} exp((4^n-1)/3x)x^n/n!.

Original entry on oeis.org

1, 1, 3, 19, 239, 6091, 305023, 30818299, 6155906879, 2484667187371, 1989929726352863, 3221489148102557179, 10362312712649347408159, 67345216546226371822133611, 869978904614825017953532433663

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..80

Cf. variants: A001831, A135753.

Flatten[{1,Table[Sum[Binomial[n,k]*((4^k-1)/3)^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 25 2013 *)

a(n)=sum(k=0,n,binomial(n,k)*((4^k-1)/3)^(n-k))

a(n)=n!*polcoeff(sum(k=0,n,exp((4^k-1)/3*x)*x^k/k!),n)

a(n) = Sum_{k=0..n} C(n,k)*[(4^k-1)/3]^(n-k).

a(n) ~ c * 2^(n^2/2+n+1/2)/(3^(n/2)*sqrt(Pi*n)), where c = Sum_{k = -infinity..infinity} 3^k*4^(-k^2) = 1.86902676808473931... if n is even and c = Sum_{k = -infinity..infinity} 3^(k+1/2)*4^(-(k+1/2)^2) = 1.87384213421283135... if n is odd. - Vaclav Kotesovec, Jun 25 2013

A360933 Expansion of e.g.f. Sum_{k>=0} exp((3^k - 1)x) x^k/k!.

Original entry on oeis.org

1, 1, 5, 37, 521, 12361, 510605, 35837677, 4348414481, 903630399121, 325415100648725, 201805338104622517, 217331913727442676761, 404193405278758441895641, 1306527408146744068362681245, 7302236837745565755664036677757

Offset: 0

Seiichi Manyama, Feb 26 2023

Cf. A001831, A360934, A360935.

Cf. A135079, A135753.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1+sum(k=1, N, exp((3^k-1)*x)*x^k/k!)))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(3^k-1)*x)^(k+1)))

a(n) = sum(k=0, n, (3^k-1)^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} x^k/(1 - (3^k - 1)*x)^(k+1).

a(n) = Sum_{k=0..n} (3^k - 1)^(n-k) * binomial(n,k).

A174122 Partial sums of A001831.

Original entry on oeis.org

1, 2, 5, 18, 105, 946, 12589, 240482, 6526289, 250119330, 13512676053, 1027978959346, 110155994874553, 16631509898085074, 3540687511804739837, 1063409634646294541250, 450894476653951603096737

Offset: 0

Jonathan Vos Post, Mar 08 2010

nonn

Partial sums of number of labeled graded partially ordered sets with n elements. The subsequence of primes in this partial sum begins: 2, 5, 12589.

Cf. A001831, A002031, A047863, A052332, A001833, A048194, A052296, A135753, A135754.

a(n) = SUM[i=0..n] A001831(i) = SUM[i=0..n] SUM[j=0..i] ((-1)^j*C(n,j)*A047863(j)).

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A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

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A135754 E.g.f.: A(x) = Sum_{n>=0} exp((4^n-1)/3*x)*x^n/n!.

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A360933 Expansion of e.g.f. Sum_{k>=0} exp((3^k - 1)*x) * x^k/k!.

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PARI

PARI

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A174122 Partial sums of A001831.

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A135754 E.g.f.: A(x) = Sum_{n>=0} exp((4^n-1)/3x)x^n/n!.

A360933 Expansion of e.g.f. Sum_{k>=0} exp((3^k - 1)x) x^k/k!.