A052852 -id:A052852 - cp's OEIS frontend

A288954 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except before the first and after the last branch node on level 0.

1, 1, 3, 13, 79, 555, 4605, 42315, 436275, 4894155, 60125625, 794437875, 11325612375, 172141044075, 2793834368325, 48009995908875, 874143494098875, 16757439016192875, 338309837281040625, 7157757510792763875, 158706419654857449375, 3673441093896736036875

Offset: 2

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels a maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the begininning and at the end. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

			See A288950 and A288953.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A288953 (variation without initial sequence).
Cf. A177145 (variation without initial and final sequence).
Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

terms = 22; egf = 1/(3(1-z))(1/Sqrt[1-z^2] + (3z^3 - z^2 - 2z + 2)/((1-z)(1-z^2))) + O[z]^terms;
CoefficientList[egf, z] Range[0, terms-1]! (* Jean-François Alcover, Dec 13 2018 *)

E.g.f.: 1/(3*(1-z))*( 1/sqrt(1-z^2) + (3*z^3-z^2-2*z+2)/((1-z)*(1-z^2)) ).

A317277 a(n) = Sum_{k=0..n} binomial(n-1,k-1)k^nn!/k!; a(0) = 1.

Original entry on oeis.org

1, 1, 6, 81, 1828, 60565, 2734926, 160109005, 11724156648, 1045312448841, 111114793839610, 13845807451708441, 1994597720747571468, 328351264019737949341, 61162428777982281583302, 12782305566531823350524805, 2975150384583838798131401296, 766253903501365584725344992529

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

nonn

a(n) is the n-th term of the Lah transform of the n-th powers.

G. C. Greubel, Table of n, a(n) for n = 0..250
N. J. A. Sloane, Transforms

Cf. A000262, A052852, A103194, A293145, A317278, A317279.

[1]cat[(&+[Binomial(n-1,j-1)*Binomial(n,j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // G. C. Greubel, Mar 09 2021

A317277:= n-> `if`(n=0,1, add(binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n)); seq(A317277(n), n=0..30); # G. C. Greubel, Mar 09 2021

Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^n n!/k!, {k, n}], {n, 17}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 - x))^k/k!, {k, n}], {x, 0, n}], {n, 17}]]

a(n) = if (n==0, 1, sum(k=0, n, binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 15 2022

[1]+[sum(binomial(n-1,j-1)*binomial(n,j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # G. C. Greubel, Mar 09 2021

a(n) = n! * [x^n] Sum_{k>=0} k^n*(x/(1 - x))^k/k!.

Name edited by Michel Marcus, Jun 15 2022

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

A009737 Expansion of e.g.f. tan(x)*exp(tan(x)).

Original entry on oeis.org

0, 1, 2, 5, 20, 81, 438, 2477, 16680, 120481, 973034, 8496245, 80252732, 817734321, 8859646110, 102873611549, 1258403748432, 16372688411713, 223202277906386, 3213260867586149, 48295209177888356, 761792907575450385, 12510350648500199814, 214507625428065409805

Offset: 0

R. H. Hardin

G. C. Greubel, Table of n, a(n) for n = 0..475

Cf. A052852, A059419, A008277.

R:=PowerSeriesRing(Rationals(), 30);
[0] cat Coefficients(R!( Laplace( Tan(x)*Exp(Tan(x)) ) )); // G. C. Greubel, Mar 09 2021

m:= 30; S:= series(tan(x)*exp(tan(x)), x, m+1); seq(j!*coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 09 2021

With[{nn=20},CoefficientList[Series[Tan[x]Exp[Tan[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 30 2011 *)

a(n):=sum((1+(-1)^(n-k))*sum(j!*stirling2(n,j)*2^(n-j-1)*(-1)^((n+k)/2+j)*binomial(j-1,k-1),j,k,n)/(k-1)!,k,1,n); /* Vladimir Kruchinin, Apr 19 2011 */

[factorial(n)*( tan(x)*exp(tan(x)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 09 2021

a(n) = Sum_{k=1..n} ((1+(-1)^(n-k))/(k-1)!) * Sum_{j=k..n} j! * Stirling2(n,j) * 2^(n-j-1)*(-1)^((n+k)/2+j)*binomial(j-1,k-1). - Vladimir Kruchinin, Apr 19 2011

a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x^2)*d/dx. Cf. A052852. a(n) = Sum_{k=1..n} k*A059419(n,k). - Peter Bala, Nov 25 2011

Extended and signs tested by Olivier Gérard, Mar 15 1997

A072678 Generalized Bell numbers B_{4,2}.

Original entry on oeis.org

1, 21, 1045, 93289, 12975561, 2581284541, 693347907421, 241253367679185, 105394372192969489, 56410454014314490981, 36271084122927079387941, 27567930377271475039277881, 24435533594428382909107147225

Offset: 1

Karol A. Penson, Jul 01 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..220
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-phys/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A090439 (alternating row sums of A090438).

f:= n -> simplify((2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1))):
map(f, [$1..30]); # Robert Israel, May 23 2016

a[n_] := n*(2n-1)!*Hypergeometric1F1[2-2n, 3, -1]; Array[a, 30] (* Jean-François Alcover, Sep 01 2016 *)

a(n) = (2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1)), n=1, 2, ... Special values of the confluent hypergeometric function 1F1.
a(n) = sum(A090438(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+(j-1)*(4-2), 2), j=1..n), k=1..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
8*n*(2*n-1)*(2*n+1)*(n+1)^2*(n+3)*(n+2)*a(n)+(2*(n+1))*(8*n^3+32*n^2+42*n+13)*a(n+1)*(n+3)*(n+2)-(8*n^2+38*n+51)*(n+3)*(n+2)*a(n+2)+(n+3)*(n+2)*a(n+3) = 0. - Robert Israel, May 23 2016
a(n) = A052852(2*n-1). - Mark van Hoeij, Sep 05 2022

Edited by Wolfdieter Lang, Dec 23 2003

A114329 Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 36, 24, 12, 0, 1, 240, 180, 60, 20, 0, 1, 1920, 1440, 540, 120, 30, 0, 1, 17640, 13440, 5040, 1260, 210, 42, 0, 1, 183120, 141120, 53760, 13440, 2520, 336, 56, 0, 1, 2116800, 1648080, 635040, 161280, 30240, 4536, 504, 72, 0, 1

Offset: 0

Vladeta Jovovic, Feb 06 2006

The average number of size 1 lists goes to 1 as n->infinity. In other words, lim_{n->infinity} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. - Geoffrey Critzer, Feb 20 2022 (after asymptotic limits by Vaclav Kotesovec given in A000262)

			Triangle begins:
    1;
    0,   1;
    2,   0,  1;
    6,   6,  0,  1;
   36,  24, 12,  0, 1;
  240, 180, 60, 20, 0, 1;
  ...

Nathaniel Johnston, Table of n, a(n) for n = 0..5150

Cf. A000262, A006152, A052845, A052852.

t:=taylor(exp(x/(1-x)+(y-1)*x),x,11):for n from 0 to 10 do for k from 0 to n do printf("%d, ",coeff(n!*coeff(t,x,n),y,k)): od: printf("\n"): od: # Nathaniel Johnston, Apr 27 2011
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=1, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 19 2022

nn = 10; Table[Take[(Range[0, nn]! CoefficientList[ Series[Exp[ x/(1 - x) - x + y x], {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}] // Grid (* Geoffrey Critzer, Feb 19 2022 *)

E.g.f.: exp(x/(1-x)+(y-1)*x). More generally, e.g.f. for number of partitions of n-set into lists with k lists of size m is exp(x/(1-x)+(y-1)*x^m).

A317365 Expansion of e.g.f. x*exp(x/(1 + x))/(1 + x).

Original entry on oeis.org

0, 1, 0, -3, 16, -75, 336, -1295, 1632, 55881, -1124000, 16722981, -229985040, 3089923837, -41225160144, 545880027225, -7069180940864, 86130735547665, -882387869940288, 3847692639294541, 171852333163131600, -8392137456287472699, 276055495385982856720, -8067943451470397940543

Offset: 0

Ilya Gutkovskiy, Jul 26 2018

sign

Inverse Lah transform of the nonnegative integers (A001477).

G. C. Greubel, Table of n, a(n) for n = 0..440
N. J. A. Sloane, Transforms
Index entries for sequences related to Laguerre polynomials

Cf. A001477, A009940, A052852.

[n eq 0 select 0 else (-1)^(n+1)*Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), 1): n in [0..25]]; // G. C. Greubel, Mar 05 2021

a:= proc(n) option remember; add((-1)^(n-k)*
      n!/(k-1)!*binomial(n-1, k-1), k=1..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

nmax = 23; CoefficientList[Series[x Exp[x/(1 + x)]/(1 + x) , {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] n!/(k - 1)!, {k, n}], {n, 0, 23}]
Join[{0}, Table[(-1)^(n+1) n! LaguerreL[n-1, 1], {n, 23}]]

a(n) = (-1)^(n+1)*n!*pollaguerre(n-1,0,1); \\ Michel Marcus, Mar 06 2021

[0 if n==0 else (-1)^(n+1)*factorial(n)*gen_laguerre(n-1, 0, 1) for n in (0..25)] # G. C. Greubel, Mar 05 2021

a(n) = Sum_{k=1..n} binomial(n-1,k-1)*n!/(k-1)!.
From G. C. Greubel, Mar 05 2021: (Start)
a(n) = n! * Hypergeometric1F1([-(n-1)], [1], -1).
a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1). (End)

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

A093190 Array t read by antidiagonals: number of {112,212}-avoiding words.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 8, 21, 16, 5, 1, 10, 39, 52, 25, 6, 1, 12, 63, 136, 105, 36, 7, 1, 14, 93, 292, 365, 186, 49, 8, 1, 16, 129, 544, 1045, 816, 301, 64, 9, 1, 18, 171, 916, 2505, 3006, 1603, 456, 81, 10, 1, 20, 219, 1432, 5225, 9276, 7315, 2864, 657, 100, 11

Offset: 1

Ralf Stephan, Apr 20 2004

t(k,n) = number of n-long k-ary words that simultaneously avoid the patterns 112 and 212.

			Square array begins as:
  1  1   1   1    1    1 ... 1*A000012;
  2  4   6   8   10   12 ... 2*A000027;
  3  9  21  39   63   93 ... 3*A002061;
  4 16  52 136  292  544 ... 4*A135859;
  5 25 105 365 1045 2505 ... ;
Antidiagonal rows begins as:
  1;
  1,  2;
  1,  4,  3;
  1,  6,  9,   4;
  1,  8, 21,  16,   5;
  1, 10, 39,  52,  25,  6;
  1, 12, 63, 136, 105, 36, 7;

G. C. Greubel, Antidiagonal rows n = 1..50, flattened
A. Burstein and T. Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.

Main diagonal is A052852.

Antidiagonal sums are in A084261 - 1.

[(&+[Factorial(j)*Binomial(k,j)*Binomial(n-k,j-1): j in [0..n-k+1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 09 2021

T[n_, k_]:= Sum[j!*Binomial[k, j]*Binomial[n-k, j-1], {j,0,n-k+1}];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

t(n,k)=sum(j=0,k,j!*binomial(k,j)*binomial(n-1,j-1))

flatten([[ sum(factorial(j)*binomial(k,j)*binomial(n-k,j-1) for j in (0..n-k+1)) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 09 2021

t(n, k) = Sum{j=0..n} j!*C(n, j)*C(k-1, j-1). (square array)

T(n, k) = Sum_{j=0..n-k+1} j!*binomial(k,j)*binomial(n-k,j-1). (number triangle) - G. C. Greubel, Mar 09 2021

A102289 Total number of odd lists in all sets of lists, cf. A000262.

Original entry on oeis.org

0, 1, 2, 15, 76, 665, 5286, 56287, 597080, 7601841, 99702730, 1484554511, 23049638052, 393702612745, 7036703742446, 135702811542495, 2737989749177776, 58848546456947297, 1321063959370833810, 31310238786268648591, 773291778432688011260, 20031956775840631151481

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A052852, A066897, A066898, A059570, A059570, A081358, A092691.

G:=(x/(1-x^2))*exp(x/(1-x)): Gser:=series(G,x=0,25): seq(n!*coeff(Gser,x^n),n=1..22); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::odd, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Rest[CoefficientList[Series[x/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
nxt[{n_,a_,b_,c_}]:={n+1,b,c,(n+1)*c+(n+1)^2*b-(n-1)^2 (n+1)*a}; NestList[ nxt,{2,0,1,2},30][[All,2]] (* Harvey P. Dale, Jan 13 2019 *)

E.g.f.: x/(1-x^2)*exp(x/(1-x)).
a(n) = n*a(n-1) + n^2*a(n-2) - (n-2)^2*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 + 7/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013

More terms from Emeric Deutsch, Jun 24 2005

a(0)=0 pepended by Alois P. Heinz, May 10 2016

cp's OEIS Frontend

A288954 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except before the first and after the last branch node on level 0.

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A317277 a(n) = Sum_{k=0..n} binomial(n-1,k-1)*k^n*n!/k!; a(0) = 1.

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

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A009737 Expansion of e.g.f. tan(x)*exp(tan(x)).

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A072678 Generalized Bell numbers B_{4,2}.

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A114329 Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.

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A317365 Expansion of e.g.f. x*exp(x/(1 + x))/(1 + x).

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A317277 a(n) = Sum_{k=0..n} binomial(n-1,k-1)k^nn!/k!; a(0) = 1.

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

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