A288268 -id:A288268 - cp's OEIS frontend

A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).

0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080

Offset: 0

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

A simple grammar.
Number of {121,212}-avoiding n-ary words of length n. - Ralf Stephan, Apr 20 2004
The infinite continued fraction (1+n)/(1+(2+n)/(2+(3+n)/(3+...))) converges to the rational number A052852(n)/A000262(n) when n is a positive integer. - David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008
a(n) is the total number of components summed over all nilpotent partial permutations of [n]. - Geoffrey Critzer, Feb 19 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Angell, A family of continued fractions, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904-911, Section 2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 820
Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.
Index entries for sequences related to Laguerre polynomials

Row sums of unsigned triangle A062139 (generalized a=2 Laguerre).
Cf. A000262. A000169, A000312.
Cf. A059114, A288268.

[n eq 0 select 0 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), -1): n in [0..30]]; // G. C. Greubel, Feb 23 2021

spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a := n -> ifelse(n = 0, 0, n!*hypergeom([-n+1], [1], -1)): seq(simplify(a(n)), n = 0..18);  # Peter Luschny, Dec 30 2024

Table[n!*SeriesCoefficient[(x/(1-x))*E^(x/(1-x)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 09 2012 *)
Table[If[n==0, 0, n!*LaguerreL[n-1, 0, -1]], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)

my(x='x+O('x^30)); concat([0], Vec(serlaplace((x/(1-x))*exp(x/(1-x))))) \\ G. C. Greubel, May 15 2018

[0 if n==0 else factorial(n)*gen_laguerre(n-1, 0, -1) for n in (0..30)] # G. C. Greubel, Feb 23 2021

D-finite with recurrence: a(1)=1, a(0)=0, (n^2+2*n)*a(n)+(-4-2*n)*a(n+1)+ a(n+2)=0.
a(n) = Sum_{m=0..n} n!*binomial(n+2, n-m)/m!. - Wolfdieter Lang, Jun 19 2001
a(n) = n*A002720(n-1). [Riordan] - Vladeta Jovovic, Mar 18 2005
Related to an n-dimensional series: for n>=1, a(n) = (n!/e)*Sum_{k_n>=k_{n-1}>=...>=k_1>=0} 1/(k_n)!. - Benoit Cloitre, Sep 30 2006
E.g.f.: (x/(1-x))*exp((x/(1-x))) = (x/(1-x))*G(0); G(k)=1+x/((2*k+1)*(1-x)-x*(1-x)*(2*k+1)/(x+(1-x)*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A000262 and A005493. - Peter Bala, Nov 25 2011
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+1/4)/sqrt(2). - Vaclav Kotesovec, Oct 09 2012
a(n) = n!*hypergeom([-n+1], [1], -1) for n>=1. - Peter Luschny, Oct 18 2014 [Simplified by Natalia L. Skirrow, 30 December 2024]
a(n) = Sum_{k=0..n} L(n,k)*k; L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014
a(n) = n!*LaguerreL(n-1, 0, -1) for n>=1. - Peter Luschny, Apr 08 2015, simplified Dec 30 2024
The series reversion of the e.g.f. equals W(x)/(1 + W(x)) = x - 2^2*x^2/2! + 3^3*x^3/3! - 4^4*x^4/4! + ..., essentially the e.g.f. for a signed version of A000312, where W(x) is Lambert's W-function (see A000169). - Peter Bala, Jun 14 2016
Equals column A059114(n, 2) for n >= 1. - G. C. Greubel, Feb 23 2021
a(n) = Sum_{k=1..n} k * A271703(n,k). - Geoffrey Critzer, Feb 19 2022

A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 13, 21, 18, 6, 73, 136, 156, 96, 24, 501, 1045, 1460, 1260, 600, 120, 4051, 9276, 15030, 16320, 11160, 4320, 720, 37633, 93289, 170142, 219450, 192360, 108360, 35280, 5040, 394353, 1047376, 2107448, 3116736, 3294480, 2405760, 1149120, 322560, 40320

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

			Triangle begins as:
    1;
    1,    1;
    3,    4,    2;
   13,   21,   18,    6;
   73,  136,  156,   96,  24;
  501, 1045, 1460, 1260, 600, 120;
  ...;
E.g.f. for T(n, 2) = (x/(1-x))^2*e^(x/(x-1)) = x^2 + 3*x^3 + 13/2*x^4 + 73/6*x^5 + 167/8*x^6 + 4051/120*x^7 + ...

Cf. A000262, A052852, A052897, A059110.

Row sums give A059115. Alternating row sums give A288268.

[Factorial(n)*Evaluate(LaguerrePolynomial(n-k, k-1), -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Table[n!*LaguerreL[n-k, k-1, -1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

T(n, k) = n! * pollaguerre(n-k, k-1, -1); \\ Michel Marcus, Feb 23 2021

flatten([[factorial(n)*gen_laguerre(n-k, k-1, -1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

E.g.f. for T(n, k) = (x/(1-x))^k * exp(x/(x-1)).
T(n, k)= Sum_{i=0..n} L'(n,i) * ( Product_{j=1..k} (i-j+1) ).
T(n, 0) = A000262(n).
T(n, 1) = A052852(n).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = n! * k! * Sum_{j=0..n} binomial(j, k)*binomial(n-1, j-1)/j!.
T(n, k) = n! * Laguerre(n-k, k-1, -1).
T(n, k) = n!*binomial(n-1, k-1)*Hypergeometric1F1([k-n], [k], -1) with T(n, 0) = Hypergeometric2F0([1-n, -n], [], 1). (End)

A361649 a(n) = (1+n)(2a(n-1) - (n-2)*a(n-2)) with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 6, 44, 380, 3768, 42112, 523072, 7141248, 106209920, 1708188416, 29525850624, 545607622144, 10730032423936, 223691600732160, 4926284479250432, 114255071320260608, 2783085758131765248, 71023717127647854592, 1894699527341113999360, 52730415074075898937344

Offset: 0

Igor Victorovich Statsenko, Mar 19 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..438
I. V. Statsenko, Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences, Applied Discrete Mathematics No. 55, Tomsk State University Publishing House, 2022, pp. 5-13.

For m=1 the formula gives the sequence A052852.

Cf. A288268, A361528.

# For recursion:
N:=20;a[0]:=1;a[1]:=1;for n from 1 to N do
a[n+1]:=(n+2)*(2*a[n]-(n-1)*a[n-1]);od;
# For closed form:
C := binomial:
a := n -> `if`(n=0, 1, add(C(n-1, i)*C(n+1, n-i)*(n-i)!*2^(i-1), i = 0..n-1)):
seq(a(n), n = 0..20);

memo=Map([0, 1; 1, 1]);
a(n)=if(mapisdefined(memo, n),mapget(memo, n), mapput(memo, n, (n+1)* (2*a(n-1) - (n-2)*a(n-2))); a(n)); \\ Winston de Greef, Mar 20 2023

a(n) = (m+n-1)*(2*a(n-1) - (n-2)*a(n-2)) where m = 2.
a(n) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(n+m-1,n-i)*(n-i)!*m^(i-1) where m = 2 for n >= 1.
E.g.f.: (1+x^2)*exp(2/(1-x))/(4*(1-x)^2*exp(2))+3/4. - Alois P. Heinz, Mar 19 2023
a(n) ~ 2^(-9/4) * exp(2*sqrt(2*n) - n - 1) * n^(n + 3/4). - Vaclav Kotesovec, Mar 20 2023

A288269 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)kx^k).

Original entry on oeis.org

1, 0, 4, 36, 336, 3840, 52800, 836640, 14864640, 291755520, 6264276480, 145962432000, 3665362821120, 98604459233280, 2827182573895680, 86016204578304000, 2766450467708928000, 93741871082943283200, 3336807307530977280000, 124443669133537276723200

Offset: 0

Seiichi Manyama, Oct 20 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..420

Cf. A052887, A288268.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(2*x^2/(1-x)^3) ))); // G. C. Greubel, Mar 10 2021

With[{m = 30}, CoefficientList[Series[Exp[2*x^2/(1-x)^3], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Mar 10 2021 *)

{a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)*k*x^k)+x*O(x^n)), n)}

[factorial(n)*( exp(2*x^2/(1-x)^3) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 10 2021

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)*k^2*a(n-k)/(n-k)! for n > 0.
E.g.f.: exp(2*x^2/(1 - x)^3). - Ilya Gutkovskiy, Jul 27 2020
a(n) ~ 3^(1/8) * exp(2/27 - (n/6)^(1/4)/12 - (n/6)^(1/2) + 8*(n/6)^(3/4) - n) * n^(n - 1/8) / 2^(7/8) * (1 - 3203/34560 * (6/n)^(1/4)). - Vaclav Kotesovec, Mar 10 2021
a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-8)*a(n-2) + 2*(n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Dec 01 2021

A361528 a(n) = (2+n)(2a(n-1) - (n-2)*a(n-2)) with a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 8, 75, 804, 9681, 129168, 1889379, 30037500, 515342817, 9484627608, 186305208219, 3888697965012, 85920579594225, 2002828537732896, 49107722192594739, 1263165207424720812, 34004577057249890241, 955970215914084949800, 28011115058953357075563, 853924857091970071203972

Offset: 0

Igor Victorovich Statsenko, Mar 23 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
I. V. Statsenko, Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences, Applied Discrete Mathematics No. 55, Tomsk State University Publishing House, 2022, pp. 5-13.

For m=1 the formula gives the sequence A052852.

Cf. A288268. For m=2 the formula gives the sequence A361649.

# For recursion:
N:=10;a[0]:=1;a[1]:=1;for n from 1 to N do
a[n+1]:=(n+3)*(2*a[n]-(n-1)*a[n-1]);od;
# For closed form:
C := binomial:
a := n -> `if`(n=0, 1, add(C(n-1, i)*C(n+2, n-i)*(n-i)!*3^(i-1), i = 0..n-1)):
seq(a(n), n = 0..20);
# Alternative:
a := n -> `if`(n=0, 1, (n + 2)!*hypergeom([1 - n], [3], -3) / 6):
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 23 2023

nmax = 20; CoefficientList[Series[23/27 + (4 + 3*x + 2*x^3)*E^(3*x/(1 - x))/(27*(1 - x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 23 2023 *)

a(n) = if(n==0, 1, my(m=3); sum(i=0, n-1, binomial(n-1, i)*binomial(n+m-1, n-i)*(n-i)!*m^(i-1))) \\ Andrew Howroyd, Mar 23 2023

a(n) = (m+n-1)*(2*a(n-1) - (n-2)*a(n-2)) where m=3, a(0)=a(1)=1.
a(n) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(n+m-1,n-i)*(n-i)!*m^(i-1) where m = 3 for n >= 1.
a(n) = (n + 2)!*hypergeom([1 - n], [3], -3) / 6 for n >= 1. - Peter Luschny, Mar 23 2023
From Vaclav Kotesovec, Mar 23 2023: (Start)
E.g.f.: 23/27 + (4 + 3*x + 2*x^3) * exp(3*x/(1-x)) / (27*(1-x)^3).
a(n) ~ exp(2*sqrt(3*n) - n - 3/2) * n^(n + 5/4) / (sqrt(2) * 3^(9/4)). (End)

Terms a(12) and beyond from Andrew Howroyd, Mar 23 2023

A347340 E.g.f.: exp( exp(exp(x) - 1) - exp(x) ).

Original entry on oeis.org

1, 0, 1, 4, 17, 91, 587, 4327, 35604, 323316, 3210600, 34574453, 400893066, 4975247460, 65755573847, 921535225267, 13643496840808, 212688569520955, 3480978391442106, 59657975022473437, 1068151956803180295, 19937983367649562025, 387243759600707804811, 7812456801157894913964

Offset: 0

Ilya Gutkovskiy, Aug 27 2021

nonn

Exponential transform of A058692.

Stirling transform of A000296.

Alois P. Heinz, Table of n, a(n) for n = 0..480

Cf. A000110, A000166, A000258, A000296, A000587, A052852, A058692, A182386, A288268.

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1), j=2..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 27 2021
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      `if`(t=0, 1, b(j, 0)-1)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021

nmax = 23; CoefficientList[Series[Exp[Exp[Exp[x] - 1] - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(exp(x)-1)-exp(x)))) \\ Michel Marcus, Aug 27 2021

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * (Bell(k) - 1) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000296(k).
a(n) = Sum_{k=0..n} binomial(n,k) * A000258(k) * A000587(n-k).

A372205 a(n) = (-1)^na((n - 2^A007814(n))/2) + a(floor((2n - 2^A007814(n))/2)) for n > 0 and a(0) = 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 5, 3, 5, 4, 4, 1, 4, 3, 7, 4, 7, 5, 10, 5, 10, 7, 15, 10, 15, 11, 5, 1, 5, 4, 9, 5, 9, 6, 13, 6, 13, 9, 20, 13, 20, 15, 17, 7, 17, 12, 27, 17, 27, 20, 37, 22, 37, 27, 52, 37, 52, 41, 6, 1, 6, 5, 11, 6, 11, 7, 16, 7, 16, 11, 25, 16, 25, 19

Offset: 0

Peter Luschny, Apr 22 2024

nonn

This sequence was originally introduced by Mikhail Kurkov in A217924 where he conjectured that A217924(n) = Sum_{k=0..2^n-1} a(k).

Cf. A007814, A000296, A217924, A288268.

f := n -> padic[ordp](n, 2):
a := proc(n) option remember; if n = 0 then return 1 fi;
(-1)^n*a((n - 2^f(n))/2) + a(floor((2*n - 2^f(n))/2)) end:
seq(a(n), n = 0..79);

Conjecture (by Mikhail Kurkov): a(2^n - 1) = A000296(n).

Conjecture (by Mikhail Kurkov): a((4^n - 1)/3) = A288268(n).

cp's OEIS Frontend

A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A361649 a(n) = (1+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0) = a(1) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A288269 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*k*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A361528 a(n) = (2+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0)=a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A347340 E.g.f.: exp( exp(exp(x) - 1) - exp(x) ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A372205 a(n) = (-1)^n*a((n - 2^A007814(n))/2) + a(floor((2*n - 2^A007814(n))/2)) for n > 0 and a(0) = 1.

Views

Author

Keywords

Comments

A361649 a(n) = (1+n)(2a(n-1) - (n-2)*a(n-2)) with a(0) = a(1) = 1.

A288269 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)kx^k).

A361528 a(n) = (2+n)(2a(n-1) - (n-2)*a(n-2)) with a(0)=a(1)=1.

A372205 a(n) = (-1)^na((n - 2^A007814(n))/2) + a(floor((2n - 2^A007814(n))/2)) for n > 0 and a(0) = 1.