A000262 -id:A000262 - cp's OEIS frontend

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

1, 1, 3, 7, 49, 321, 2131, 19783, 195777, 2101249, 25721731, 340358151, 4902173233, 75688032577, 1253701725459, 22347046050631, 418439924732161, 8318748086461953, 175769214730290307, 3871849719998940679, 89734800330818444721, 2187944831367633226561

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A000726, A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := x*(1+x)/(1-x^3) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(0=
      irem(j, 3), 0, a(n-j)*j!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

CoefficientList[Series[E^(x*(1+x)/(1-x^3)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: exp(x*(1+x)/(1-x^3)).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + 2*(n-3)*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 6^(-1/4) * n^(n-1/4) * exp(2/3*sqrt(6*n)-n) * (1 - 43/(48*sqrt(6*n))). - Vaclav Kotesovec, Sep 25 2013

2 more terms from R. J. Mathar, Feb 06 2008

A096965 Number of sets of even number of even lists, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 37, 241, 2101, 18271, 201097, 2270017, 29668681, 410815351, 6238931821, 101560835377, 1765092183037, 32838929702671, 644215775792401, 13441862819232001, 293976795292186897, 6788407001443004647, 163735077313046119861, 4142654439686285737201

Offset: 0

Vladeta Jovovic, Aug 18 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000262, A088026, A088009, A096939, A349776.

a:= proc(n) option remember;  `if`(n<4, [1$3, 7][n+1], ((2*n-3)
      *a(n-1)+(n-1)*(2*n^2-8*n+7)*a(n-2) + (n-2)*(n-1)*(2*n-5)
      *a(n-3)-(n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4))/(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 01 2021

Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))]Cosh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* Robert G. Wilson v, Aug 19 2004 *)

E.g.f.: exp(x/(1-x^2))*cosh(x^2/(1-x^2)).
a(n) = (n!*sum(m=floor((n+1)/2)..n, (binomial(n-1,2*m-n-1))/(2*m-n)!)). - Vladimir Kruchinin, Mar 10 2013
Recurrence: (n-2)*a(n) = (2*n-3)*a(n-1) + (n-1)*(2*n^2 - 8*n + 7)*a(n-2) + (n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/(2*sqrt(2)) * (1-5/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
From Alois P. Heinz, Dec 01 2021: (Start)
a(n) = A000262(n) - A096939(n).
a(n) = |Sum_{k=0..n} (-1)^k * A349776(n,k)|. (End)

More terms from Robert G. Wilson v, Aug 19 2004

a(0)=1 prepended by Alois P. Heinz, Dec 01 2021

A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.

Original entry on oeis.org

0, 1, 5, 31, 217, 1781, 16501, 172915, 1998641, 25468777, 352751941, 5292123431, 85297925065, 1472161501981, 27039872306357, 527253067633531, 10865963240550241, 236088078855319505, 5390956470528548101, 129102989125943058607, 3234053809095307670201, 84596120521251178630981, 2305894874979300173268085

Offset: 0

Vladeta Jovovic, Jul 27 2004

			For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(4)= 24*4+24*3+12*2+12*2+1*1 = 217.

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

Cf. A028417, A028418, A046746, A006128, A097145, A097147, A097148.

b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
      b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

b[n_, m_] := b[n, m] = If[n == 0, m, Sum[j! b[n-j, Max[m, j]] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

N=50; x='x+O('x^N);
egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) );
Vec( serlaplace(egf) ) /* show terms */

E.g.f.: exp(x/(1-x))*Sum_{k>0} (1-exp(x^k/(x-1))).

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

A113236 Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262.

Original entry on oeis.org

1, 1, 3, 7, 49, 321, 2851, 24823, 256257, 2887489, 36759331, 507010791, 7597222513, 122184356737, 2106356007939, 38693238713431, 754792977928321, 15572911248409473, 338800604611562947, 7749991799652960199, 185934065196259734321, 4667877395135551746241

Offset: 0

Karol A. Penson, Oct 19 2005

nonn

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

Cf. A052845, A113235.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(1-x^2+x^3)/(1-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=3, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

Range[0, 18]!*CoefficientList[ Series[ Exp[x*(1-x^2+x^3)/(1 - x)], {x, 0, 18}], x] (* Zerinvary Lajos, Mar 23 2007 *)
a[n_] := a[n] = If[n==0, 1, Sum[If[j==3, 0, a[n-j]*Binomial[n-1, j-1]*j!], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 11 2017, after Alois P. Heinz *)

x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ G. C. Greubel, May 17 2018

E.g.f.: exp(x*(1-x^2+x^3)/(1-x)).
Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 3*k, -1, -1]/k!, {k, 0, Floor[n/3]}], n=0, 1....
a(n) ~ exp(-3/2+2*sqrt(n)-n)*n^(n-1/4)/sqrt(2). - Vaclav Kotesovec, Jun 22 2013

A133289 Riordan matrix T from A084358 (lists of sets of lists) inverse to the Riordan matrix TI = 2I-A129652 formed from A000262 (number of sets of lists) and reciprocal under a partition transform.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 37, 15, 3, 1, 363, 148, 30, 4, 1, 4441, 1815, 370, 50, 5, 1, 65133, 26646, 5445, 740, 75, 6, 1, 1114009, 455931, 93261, 12705, 1295, 105, 7, 1, 21771851, 8912072, 1823724, 248696, 25410, 2072, 140, 8, 1

Offset: 0

Tom Copeland, Oct 16 2007, Nov 30 2007

T(n,k) is simply constructed from Pascal's triangle PT and A084358 through multiplication along the diagonals. Taking the matrix inverse gives TI = 2I-A129652 = PT times diagonal multiplication by -A000262 with the sign of the first term flipped to positive.

T and TI are also reciprocals under the list partition transform described in A133314.

			Triangle starts:
1,
1, 1,
5, 2, 1,
37, 15, 3, 1,
363, 148, 30, 4, 1,
4441, 1815, 370, 50, 5, 1,
...

Vincenzo Librandi, Rows n = 0..100, flattened
T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7

Cf. A131202.

max = 7; s = Series[Exp[x*t]/(2-Exp[x/(1-x)]), {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]*n!; t[0, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 23 2014 *)

T(n,k) = binomial(n,k) * A084358(n-k).

E.g.f.: exp(xt) / { 2 - exp[x/(1-x)] }.

A096939 Number of sets of odd number of even lists, cf. A000262.

Original entry on oeis.org

0, 2, 6, 36, 260, 1950, 19362, 193256, 2326536, 29272410, 413257790, 6231230412, 101415565836, 1769925341366, 32734873484250, 646218442877520, 13404753632014352, 294656673023216946, 6775966692145553526

Offset: 1

Vladeta Jovovic, Aug 18 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000262, A088026, A088009, A096965.

Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))] Sinh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* Robert G. Wilson v, Aug 19 2004 *)

E.g.f.: exp(x/(1-x^2))*sinh(x^2/(1-x^2)).
Recurrence: (n-2)*a(n) = (2*n-3)*a(n-1) + (n-1)*(2*n^2 - 8*n + 7)*a(n-2) + (n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/(2*sqrt(2)) * (1-5/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = A000262(n) - A096965(n). - Alois P. Heinz, Dec 01 2021

More terms from Robert G. Wilson v, Aug 19 2004

A102760 Number of partitions of n-set into "lists", in which every even list appears an even number of times, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 37, 241, 1381, 13231, 140617, 1483777, 16211881, 217551511, 3384215341, 50221272817, 782154787597, 13913712591871, 272739557719441, 5282625708305281, 106588332600443857, 2354480141600267047, 56238135934525073461, 1338131691952924913521

Offset: 0

Vladeta Jovovic, Feb 10 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A003483, A006950, A052565, A088026.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i::even and j::odd, 0, b(n-i*j, i-1)*
      multinomial(n, n-i*j, i$j)/j!*i!^j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[EvenQ[i] && OddQ[j], 0, b[n-i*j, i- 1] * multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!*i!^j], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

E.g.f.: exp(x/(1-x^2))*Product_{k>0} cosh(x^(2*k)).

a(0)=1 prepended by Alois P. Heinz, May 10 2016

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).

A131202 A coefficient tree from the list partition transform relating A111884, A084358, A000262, A094587, A128229 and A131758.

Original entry on oeis.org

1, -1, 3, 1, -8, 13, 1, 11, -61, 73, -19, 66, 66, -494, 501, 151, -993, 2102, -298, -4293, 4051, -1091, 9528, -33249, 52816, -21069, -39528, 37633, 7841, -82857, 378261, -929101, 1207299, -560187, -375289, 394353, -56519, 692422, -3832928, 12255802, -23834210, 26643994, -12620672, -3481562, 4596553

Offset: 1

Tom Copeland, Oct 22 2007, Nov 30 2007

Construct the infinite array of polynomials
a(0,t) =   1
a(1,t) =   1
a(2,t) =  -1 +   3*t
a(3,t) =   1 -   8*t +   13*t^2
a(4,t) =   1 +  11*t -   61*t^2 +  73*t^3
a(5,t) = -19 +  66*t +   66*t^2 - 494*t^3 +  501*t^4
a(6,t) = 151 - 993*t + 2102*t^2 - 298*t^3 - 4293*t^4 + 4051*t^5
This array is the reciprocal array of the following array b(n,t) under the list partition transform and its associated operations described in A133314.
b(0,t) = 1 and b(n,t) = -A000262(n)*(t-1)^(n-1) for n > 0.
Then A111884(n) = a(n,0).
Lower triangular matrix A094587 = binomial(n,k)*a(n-k,1).
A084358(n) = a(n,2).
Signed A128229 = matrix inverse of binomial(n,k)*a(n-k,1) = binomial(n,k)*b(n-k,1) = A132013.
As t tends to infinity, a(n,t)/t^(n-1) tends to A000262(n) for n > 0.
The P(n,t) of A131758 can be constructed from T(n,k,t) = binomial(n,k)*a(n-k,t) by letting T(n,k,t) multiply the column vector c(n,t) given by c(0,t) = 0! and c(n,t) = n!*(t-1)^(n-1) for n > 0. The P(n,t) have rich associations to other sequences.

CoefficientList[#, t] & /@ (# Range@Length@#!) &@ Rest@CoefficientList[(t-1) / (t - Exp[x(t-1)/(1-x(t-1))]) + O[x]^10 // Simplify, x] // Flatten (* Andrey Zabolotskiy, Feb 19 2024 *)

T(n) = [Vecrev(p) | p<-Vec(-1 + serlaplace((y-1) / (y - exp(x*(y-1)/(1-x*(y-1)) + O(x*x^n) ))))]
{ my(A=T(7)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 19 2024

E.g.f. for the row polynomials, which are a(n, t) for n > 0, is:
(t-1) / (t - exp(x*(t-1)/(1-x*(t-1)))).
E.g.f. for the polynomials b(n, t), introduced above, is the reciprocal of that.

Rows 7-9 added and offset changed by Andrey Zabolotskiy, Feb 19 2024

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

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

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A096965 Number of sets of even number of even lists, cf. A000262.

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A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.

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A113236 Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262.

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A133289 Riordan matrix T from A084358 (lists of sets of lists) inverse to the Riordan matrix TI = 2I-A129652 formed from A000262 (number of sets of lists) and reciprocal under a partition transform.

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A096939 Number of sets of odd number of even lists, cf. A000262.

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A102760 Number of partitions of n-set into "lists", in which every even list appears an even number of times, cf. A000262.

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