A000262 -id:A000262 - cp's OEIS frontend

A102290 Total number of even lists in all sets of lists, cf. A000262.

0, 0, 2, 6, 60, 380, 3990, 37002, 450296, 5373720, 76018410, 1096730030, 17814654132, 299645294676, 5511836578430, 105550556136690, 2171244984679920, 46545825736022192, 1059273836225051346, 25100215228045842390, 626204775725372971820, 16239127347086448236460

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

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

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[0,0]cat[Factorial(n)*(&+[(-1)^(n+j)*l(j,-1): j in [0..n-2]]): n in [2..30]]; // G. C. Greubel, Mar 09 2021

Gser:=series(x^2*exp(x/(1-x))/(1-x^2),x=0,22):seq(n!*coeff(Gser,x^n),n=1..21); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::even, [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^2/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
Table[If[n<2, 0, n!*Sum[(-1)^(n-j)*LaguerreL[j, -1], {j,0,n-2}]], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[0,0]+[factorial(n)*sum((-1)^(n+j)*gen_laguerre(j,0,-1) for j in (0..n-2)) for n in (2..30)] # G. C. Greubel, Mar 09 2021

E.g.f.: x^2/(1-x^2)*exp(x/(1-x)).
Recurrence: (n-2)*a(n) = (n-2)*n*a(n-1) + (n-1)^2*n*a(n-2) - (n-3)*(n-2)*(n-1)*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 - 41/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = n! * Sum_{j=0..n-2} (-1)^(n+j)*LaguerreL(j, -1) for n>1 with a(0)=a(1)=0. - G. C. Greubel, Mar 09 2021

More terms from Emeric Deutsch, Mar 27 2005

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

A173227 Partial sums of A000262.

Original entry on oeis.org

1, 2, 5, 18, 91, 592, 4643, 42276, 436629, 5033182, 63974273, 888047414, 13358209647, 216334610860, 3751352135263, 69325155322184, 1359759373992105, 28206375825238458, 616839844140642301, 14181213537729200474, 341879141423814854915, 8623032181189674581256

Offset: 0

Jonathan Vos Post, Feb 13 2010

nonn

Partial sums of the number of "sets of lists": number of partitions of {1,..,n} into any number of lists, where a list means an ordered subset. The subsequence of primes begins: 2, 5, 4643, 616839844140642301.

			a(20) = 1 + 1 + 3 + 13 + 73 + 501 + 4051 + 37633 + 394353 + 4596553 + 58941091 + 824073141 + 12470162233 + 202976401213 + 3535017524403 + 65573803186921 + 1290434218669921 + 26846616451246353 + 588633468315403843 + 13564373693588558173 + 327697927886085654441.

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A001263, A001700, A002868, A008297, A052852, A066668, A111596,.

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[n eq 0 select 1 else 1 + (&+[ Factorial(j)*( l(j,-1) - l(j-1,-1) ): j in [1..n]]): n in [0..25]]; // G. C. Greubel, Mar 09 2021

b:= proc(n) option remember; `if`(n=0, 1, add(
       b(n-j)*j!*binomial(n-1, j-1), j=1..n))
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

With[{m = 25}, CoefficientList[Exp[x/(1-x)] + O[x]^m, x] Range[0, m-1]!// Accumulate] (* Jean-François Alcover, Nov 21 2020 *)
Table[1 +Sum[j!*(LaguerreL[j, -1] -LaguerreL[j-1, -1]), {j,n}], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

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

From Vaclav Kotesovec, Oct 25 2016: (Start)
a(n) = 2*n*a(n-1) - (n^2 - n + 1)*a(n-2) + (n-2)*(n-1)*a(n-3).
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/sqrt(2) * (1 - 5/(48*sqrt(n))).
(End)
a(n) = 1 + Sum_{j=1..n} j!*( LaguerreL(j,-1) - LaguerreL(j-1,-1) ). - G. C. Greubel, Mar 09 2021

A192681 Floor-sqrt transform of Lah partition numbers (A000262).

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 63, 193, 627, 2143, 7677, 28706, 111669, 450529, 1880164, 8097765, 35922614, 163849371, 767224522, 3682984346, 18102428784, 91000840873, 467393250911, 2450438244585, 13102651355735, 71398380128514, 396202573696587

Offset: 0

Emanuele Munarini, Jul 07 2011

nonn

A000262 := proc(n) option remember: if n=0 then RETURN(1) fi: if n=1 then RETURN(1) fi: (2*n-1)*procname(n-1) - (n-1)*(n-2)*procname(n-2) end proc:
A192681 := proc(n) floor(sqrt(A000262(n))) ; end proc: # R. J. Mathar, Jul 13 2011

FSFromExpSeries[f_,x_,n_] := Map[Floor[Sqrt[#]]&,CoefficientList[Series[f,{x,0,n}],x]Table[k!,{k,0,n}]]
FSFromExpSeries[Exp[x/(1-x)],x,60]

a(n) = floor(sqrt(A000262(n))).

All terms replaced by R. J. Mathar, Jul 13 2011

A276960 a(n) = A000262(n)^2.

Original entry on oeis.org

1, 1, 9, 169, 5329, 251001, 16410601, 1416242689, 155514288609, 21128299481809, 3474052208270281, 679096541717605881, 155504946117339546289, 41199419449380747871369, 12496348897836314700506409

Offset: 0

Emanuele Munarini, Sep 27 2016

nonn

Cf. A000262, A105278, A008297, A276964.

Table[HypergeometricPFQ[{-n+1,-n},{},1]^2,{n,0,100}]

makelist(hypergeometric([-n+1,-n],[],1)^2,n,0,12);

Recurrence: (2*n+3)*a(n+3)-(2*n+5)*(3*n^2+13*n+13)*a(n+2)+(n+2)*(n+1)*(2*n+3)*(3*n^2+13*n+13)*a(n+1)-n^2*(n+1)^3*(n+2)*(2 n+5)*a(n) = 0.

Asymptotic: a(n) ~ exp(-2*n+4*sqrt(n)-1)*n^(2*n-1/2)/2 * (1 - 5/(24*sqrt(n)) - 35/(1152*n)).

A276964 a(n) = A000262(n)*A000262(n+1).

Original entry on oeis.org

1, 3, 39, 949, 36573, 2029551, 152451283, 14840686449, 1812664465209, 270925848659323, 48571769994336831, 10276325760127883853, 2531148652596607988629, 717525135328209346300839, 231804543407519025287933163

Offset: 0

Emanuele Munarini, Sep 27 2016

nonn

Cf. A000262, A105278, A008297, A276960.

Table[HypergeometricPFQ[{-n+1,-n},{},1]HypergeometricPFQ[{-n,-n-1},{},1],{n,0,100}]

makelist(hypergeometric([-n+1,-n],[],1)*hypergeometric([-n,-n-1],[],1),n,0,12);

Recurrence: (2*n+3)*a(n+3)-(2*n+3)*(3*n^2+19*n+29)*a(n+2)+(n+2)*(n+1)*(2*n+7)*(3*n^2+13*n+13)*a(n+1)-n*(n+1)^3*(n+2)^2*(2*n+7)*a(n) = 0.
Asymptotic: a(n) ~ (1/2)*exp(-2*n+2*sqrt(n)+2*sqrt(n+1)-1)*(475/(110592*n^(3/2))+9025/(21233664*n^2)-5/(24*sqrt(n))-35/(1152*n)+1)*n^(2*n+1/2).
a(n) ~ exp(-1 + 4*sqrt(n) - 2*n) * n^(2*n + 1/2)/2 * (1 + 19/(24*sqrt(n)) + 589/(1152*n)). - Vaclav Kotesovec, Sep 27 2016

A351823 Triangular array read by rows. T(n,k) is the number of sets of lists (as in A000262(n)) with exactly k size 2 lists, n >= 0, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 1, 2, 7, 6, 49, 12, 12, 301, 140, 60, 2281, 1470, 180, 120, 21211, 12642, 2940, 840, 220417, 127736, 41160, 3360, 1680, 2528569, 1527192, 455112, 70560, 15120, 32014801, 19837530, 5748120, 1234800, 75600, 30240, 442974511, 278142590, 83995560, 16687440, 1940400, 332640

Offset: 0

Geoffrey Critzer, Feb 20 2022

From the asymptotic estimate of A000262(n) provided by Vaclav Kotesovec we deduce that in the limit as n gets big the average number of size 2 lists is equal to 1. In other words, lim_{n->oo} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. Generally for any j >= 1, the average number of size j lists equals 1 in the limit as n -> oo.

			Triangle T(n,k) begins:
      1;
      1;
      1,     2;
      7,     6;
     49,    12,   12;
    301,   140,   60;
   2281,  1470,  180, 120;
  21211, 12642, 2940, 840;
  ...

Column k=1 gives A113235.
T(n,floor(n/2)) gives A081125.
T(2n,n) gives A001813.
Cf. A000262 (row sums) A006152, A114329, A351825.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=2, 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/2))(b(n)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 20 2022

nn = 7; Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Exp[ x/(1 - x) - x ^2 + y x^2], {x, 0, nn}], {x, y}]] // Grid

E.g.f.: exp(x/(1-x) - x^2 + y*x^2).

Sum_{k=0..floor(n/2)} k * T(n,k) = A351825(n). - Alois P. Heinz, Feb 24 2022

A351825 Total number of size 2 lists in all sets of lists partitioning [n] (cf. A000262).

Original entry on oeis.org

0, 0, 2, 6, 36, 260, 2190, 21042, 226856, 2709576, 35491770, 505620830, 7780224012, 128555409996, 2269569526406, 42625044254730, 848404205856720, 17836074466842512, 394872870912995826, 9181542826326252726, 223680717959853460340, 5697036951307194432660, 151396442683371572351742

Offset: 0

Geoffrey Critzer, Feb 20 2022

nonn

Cf. A000262, A006152, A129652, A351823.

nn = 22; Range[0, nn]! CoefficientList[Series[D[Exp[ x/(1 - x) - x ^2 + y x^2], y] /. y -> 1, {x, 0, nn}], x]
Join[{0, 0, 2}, Table[n!*Hypergeometric1F1[n-1, 2, 1]/E, {n, 3, 25}]] (* Vaclav Kotesovec, Feb 21 2022 *)

a(n) = 2*binomial(n,2)*A000262(n-2).
E.g.f.: x^2*exp(x/(1-x)) = d/dy G(x,y)|y=1 where G(x,y) is the e.g.f. for A351823.
a(n) = Sum_{k=0..floor(n/2)} k * A351823(n,k).
a(n) ~ n^(n - 1/4) * exp(2*sqrt(n) - n - 1/2) / sqrt(2) * (1 - 101/(48*sqrt(n))). - Vaclav Kotesovec, Feb 21 2022
a(n) = 2 * A129652(n,2). - Alois P. Heinz, Feb 22 2022
Recurrence: (n-2)*a(n) = n*(2*n-5)*a(n-1) - (n-4)*(n-1)*n*a(n-2). - Vaclav Kotesovec, Mar 20 2023

A373425 a(n) = A000111(n) * A000262(n). Row sums of A373426.

Original entry on oeis.org

1, 1, 3, 26, 365, 8016, 247111, 10236176, 546178905, 36478244608, 2977762858411, 291550484700672, 33703918027674245, 4540228104291094528, 704744561517173519343, 124836607292749756516352, 25023470823661358817690545, 5634174369285939855014166528, 1415592664236058550974684119763

Offset: 0

Peter Luschny, Jun 07 2024

nonn

Cf. A000111, A000262, A373426.

A265608 Triangle read by rows: Bell transform of A000262.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 13, 15, 6, 1, 0, 73, 95, 45, 10, 1, 0, 501, 723, 390, 105, 15, 1, 0, 4051, 6405, 3843, 1190, 210, 21, 1, 0, 37633, 64615, 42462, 14693, 3010, 378, 28, 1, 0, 394353, 730359, 519993, 197442, 45423, 6678, 630, 36, 1

Offset: 0

Peter Luschny, Dec 20 2015

See A264428 and the link for the definition of the Bell transform.

			[n\k 0      1      2      3      4     5    6   7  8]
[0] [1]
[1] [0,     1]
[2] [0,     1,     1]
[3] [0,     3,     3,     1]
[4] [0,    13,    15,     6,     1]
[5] [0,    73,    95,    45,    10,    1]
[6] [0,   501,   723,   390,   105,   15,   1]
[7] [0,  4051,  6405,  3843,  1190,  210,  21,  1]
[8] [0, 37633, 64615, 42462, 14693, 3010, 378, 28, 1]

Peter Luschny, The Bell transform

Cf. A000262, A264428.

# uses[bell_transform from A264428]
def A265608_row(n):
    fact = [factorial(k) for k in (1..n)]
    fact2 = [sum(bell_transform(k, fact)) for k in range(n)]
    return bell_transform(n, fact2)
[A265608_row(n) for n in (0..9)]

A349780 Triangle read by rows, T(n, k) = A000262(n) - A349776(n, n - k) for n > 0 and T(0, 0) = 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 7, 13, 0, 1, 13, 49, 73, 0, 1, 21, 141, 381, 501, 0, 1, 31, 331, 1531, 3331, 4051, 0, 1, 43, 673, 4873, 17473, 32593, 37633, 0, 1, 57, 1233, 12993, 71793, 212913, 354033, 394353, 0, 1, 73, 2089, 30313, 241993, 1088713, 2782153, 4233673, 4596553

Offset: 0

Peter Luschny, Nov 30 2021

			[0] 1;
[1] 0, 1;
[2] 0, 1,  3;
[3] 0, 1,  7,   13;
[4] 0, 1, 13,   49,    73;
[5] 0, 1, 21,  141,   381,    501;
[6] 0, 1, 31,  331,  1531,   3331,    4051;
[7] 0, 1, 43,  673,  4873,  17473,   32593,   37633;
[8] 0, 1, 57, 1233, 12993,  71793,  212913,  354033, 394353;
[9] 0, 1, 73, 2089, 30313, 241993, 1088713, 2782153, 4233673, 4596553.

Row sums are A052852 for n >= 1.

Cf. A000262, A349776.

cp's OEIS Frontend

A102290 Total number of even lists in all sets of lists, cf. A000262.

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A173227 Partial sums of A000262.

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A192681 Floor-sqrt transform of Lah partition numbers (A000262).

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A276960 a(n) = A000262(n)^2.

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A276964 a(n) = A000262(n)*A000262(n+1).

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A351823 Triangular array read by rows. T(n,k) is the number of sets of lists (as in A000262(n)) with exactly k size 2 lists, n >= 0, 0 <= k <= floor(n/2).

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Maple

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A351825 Total number of size 2 lists in all sets of lists partitioning [n] (cf. A000262).

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A373425 a(n) = A000111(n) * A000262(n). Row sums of A373426.

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A265608 Triangle read by rows: Bell transform of A000262.