A367847 -id:A367847 - cp's OEIS frontend

A052820 Expansion of e.g.f. 1/(1 - x + log(1 - x)).

1, 2, 9, 62, 572, 6604, 91526, 1480044, 27353448, 568731648, 13138994112, 333895239072, 9256507508112, 278000959058016, 8991458660924112, 311585506208924064, 11517363473843526912, 452332548042633835776

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

a(n) is the number of ways to seat n people at circular tables, then linearly order the tables, then designate some (possibly all or none) of the tables at which only one person is seated. a(2) = 9 because we have: (1)(2), (1')(2), (1)(2'), (1')(2'), (2)(1), (2')(1), (2)(1'), (2')(1'), (1,2). Cf. A007840. - Geoffrey Critzer, Nov 05 2013

Seiichi Manyama, Table of n, a(n) for n = 0..395
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 Mar 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 785
Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Cascaded Analytic Nonlinear Systems, 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, December 12-15, 2011, pp. 3830-3835.
M. Thitsa and W. S. Gray, On the radius of convergence of cascaded analytic nonlinear systems: The SISO case, System Theory (SSST), 2011 IEEE 43rd Southeastern Symposium on, 14-16 March 2011, pp. 30-36.
Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, 2012, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012

Cf. A030178.
Cf. A367851, A367852, A367853.
Cf. A367845, A367846, A367847.

spec := [S,{C=Cycle(Z),B=Union(C,Z),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/(1-x+Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

E.g.f.: -1/(-1+x+log(-1/(-1+x))).
a(n) ~ n! * (1/(1-LambertW(1)))^n/(1/LambertW(1)-LambertW(1)). - Vaclav Kotesovec, Oct 01 2013
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013

A367845 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)).

Original entry on oeis.org

1, 3, 22, 250, 3816, 72968, 1675568, 44901456, 1375306368, 47392683648, 1814635323648, 76430014409472, 3511792144942080, 174806087920727040, 9370642040786049024, 538202280800536799232, 32972397141008692445184, 2146270648672407967137792

Offset: 0

Seiichi Manyama, Dec 02 2023

nonn

Cf. A052820, A367846, A367847.

Cf. A367835, A367828.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

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

a(n) ~ n! * 2^(n+1) / ((1/LambertW(1/(2*exp(1/2))) - 1 - 2*LambertW(1/(2*exp(1/2)))) * (1 - 2*LambertW(1/(2*exp(1/2))))^n). - Vaclav Kotesovec, Dec 02 2023

A367846 Expansion of e.g.f. 1/(1 - x + log(1 - 3*x)).

Original entry on oeis.org

1, 4, 41, 654, 14028, 377112, 12177126, 458916588, 19769059944, 958125646080, 51597765220608, 3056601306206016, 197532472461453072, 13829353660386169344, 1042679226974498229456, 84229294995413626072608, 7257792124889497549663488

Offset: 0

Seiichi Manyama, Dec 02 2023

nonn

Cf. A052820, A367845, A367847.

Cf. A367836.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

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

A367853 Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)/4).

Original entry on oeis.org

1, 2, 12, 128, 1952, 38464, 926336, 26323968, 861419520, 31882358784, 1316275003392, 59954841649152, 2985997926727680, 161401148097036288, 9408988894966579200, 588381964243109412864, 39285329204482179858432, 2789234068575581984784384

Offset: 0

Seiichi Manyama, Dec 02 2023

nonn

Cf. A052820, A367851, A367852.

Cf. A352071, A355112, A367847.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 4^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 4^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

cp's OEIS Frontend

A052820 Expansion of e.g.f. 1/(1 - x + log(1 - x)).

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A367845 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)).

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A367846 Expansion of e.g.f. 1/(1 - x + log(1 - 3*x)).

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A367853 Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)/4).

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