A000248 -id:A000248 - cp's OEIS frontend

A209801 The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^2 in the equality sense.

1, 2, 7, 30, 152, 878, 5653, 39952, 306419, 2527984, 22277080, 208483014, 2062199125, 21472152822, 234526948183, 2678973711602, 31919113081724, 395750219427590, 5095324584255641, 67996852799627404, 938939425151949211, 13395286474394627364, 197162835188949226772

Offset: 0

Adam Goyt, Mar 13 2012

nonn

A partition of the set [n] is a family nonempty disjoint sets whose union is [n]. The blocks are written in order of increasing minima. A partition of the set [n] can be written as a word p=p_1p_2...p_n where p_i=j if element i is in block j. A partition q=q_1q_2...q_n contains partition p=p_1p_2...p_k if there is a subword q_{i_1}q_{i_2}...q_{i_k} such that q_{i_a}

			For n=2 the a(2)=7 solutions are 1^11^1, 1^21^1, 1^21^2, 1^12^1, 1^12^2, 1^22^1, 1^22^2.

Alois P. Heinz, Table of n, a(n) for n = 0..535
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012

Cf. A000110, A000248.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(j+1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 29 2019

Table[Sum[BellY[n, k, Range[n] + 1], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

{a(n)=n!*polcoeff(exp((1+x)*exp(x +x*O(x^n))-1),n)} \\ Paul D. Hanna, Jun 11 2012

E.g.f.: exp( (1+x)*exp(x) - 1 ). - Paul D. Hanna, Jun 11 2012
a(n) = Sum_{i=0..n} Sum_{j=0..floor((n-i)/2)} binomial(n, i)*binomial(n-i, j)*(Sum_{p=j..n-i-j} binomial(n-i-j, p)*S(p, j)*j!*B(n-i-j-p)), where B(n) is the n-th Bell number and S(n,k) is the Stirling number of the second kind.
a(n) = Sum_{j=1..n} (j+1) * binomial(n-1,j-1) * a(n-j) for n>0, a(0)=1. - Alois P. Heinz, Aug 29 2019

A292978 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 2, 10, 15, 1, 0, 0, 6, 41, 52, 1, 0, 0, 6, 24, 196, 203, 1, 0, 0, 0, 24, 140, 1057, 877, 1, 0, 0, 0, 24, 60, 870, 6322, 4140, 1, 0, 0, 0, 0, 120, 480, 5922, 41393, 21147, 1, 0, 0, 0, 0, 120, 360, 5250, 45416, 293608, 115975

Offset: 0

Seiichi Manyama, Sep 27 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   1,  1,  0,  0,  0, ...
   2,  3,  2,  0,  0, ...
   5, 10,  6,  6,  0, ...
  15, 41, 24, 24, 24, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A000110, A000248, A216507, A292889, A292979.
Rows n=0 gives A000012.
Main diagonal gives A000142.
Cf. A292973.

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << f(k) * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292978(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A292978(20)

T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(j! * (n-k*j)!) for k > 0. - Seiichi Manyama, Jul 10 2022

A354550 Expansion of e.g.f. exp( x * exp(x^2/2) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 46, 241, 1156, 6889, 44668, 300241, 2328976, 18390901, 159273544, 1461200833, 13995753136, 144068872081, 1531949061136, 17259159775969, 202543867724608, 2474236899786781, 31633380519660256, 417760492214548561, 5751414293905728064

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..518

Cf. A000248, A354551, A354552.

Cf. A216688.

With[{nn=30},CoefficientList[Series[Exp[x Exp[x^2/2]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 03 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^2/2)))))

a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(2^k*k!*(n-2*k)!));

a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(2^k * k! * (n - 2*k)!).

A354551 Expansion of e.g.f. exp( x * exp(x^3/6) ).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 211, 1401, 8065, 37241, 240021, 1997821, 13856701, 94418325, 874328911, 8304303281, 69158458881, 658339599601, 7454839614985, 78224066633781, 805961931388741, 9828080719704941, 124199805022959051, 1466207770078872745

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

This sequence is different from A143567.

Seiichi Manyama, Table of n, a(n) for n = 0..523

Cf. A000248, A354550, A354552.

With[{nn=30},CoefficientList[Series[Exp[x*Exp[x^3/6]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 03 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^3/6)))))

a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(6^k*k!*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(6^k * k! * (n - 3*k)!).

A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 24, 12, 1, 5, 80, 90, 20, 1, 6, 240, 540, 240, 30, 1, 7, 672, 2835, 2240, 525, 42, 1, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 10, 11520, 262440, 860160, 787500, 272160, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. - Peter Luschny, Mar 13 2009
T(n,k) is the sum of the products of the cardinality of the blocks (cells) in the set partitions of {1,2,..,n} into exactly k blocks.
From Peter Bala, Jul 22 2014: (Start)
Exponential Riordan array [(1+x)*exp(x), x*exp(x)].
Let M = A093375, the exponential Riordan array [(1+x)*exp(x), x], and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
  /I_k 0\
  \ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... - see the Example section. (End)
The Bell transform of n+1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins
1;
2, 1;
3, 6, 1;
4, 24, 12, 1; ...
From _Peter Bala_, Jul 22 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1          \/1        \/1        \      /1           \
|2  1       ||0 1      ||0 1      |      |2  1        |
|3  4  1    ||0 2 1    ||0 0 1    |... = |3  6  1     |
|4  9  6 1  ||0 3 4 1  ||0 0 2 1  |      |4 24 12  1  |
|5 16 18 8 1||0 4 9 6 1||0 0 3 4 1|      |5 80 90 20 1|
|...        ||...      ||...      |      |...         | (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Idempotent Number

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248. A093375.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n+1,k+1)*(k+1)^(n-k): seq(seq(T(n, k), k=0..n), n=0..10); # Georg Fischer, Oct 27 2021

t = Transpose[ Table[ Range[0, 11]! CoefficientList[ Series[(x Exp[x])^n/n!, {x, 0, 11}], x], {n, 11}]]; Table[ t[[n, k]], {n, 2, 11}, {k, n - 1}] // Flatten (* or simply *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Table[t[n, k], {n, 10}, {k, n}] // Flatten

for(n=1, 25, for(k=1, n, print1(binomial(n,k)*k^(n-k), ", "))) \\ G. C. Greubel, Jan 05 2017

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: n+1, 10) # Peter Luschny, Jan 18 2016

A133399 Triangle T(n,k)=number of forests of labeled rooted trees with n nodes, containing exactly k trees of height one, all others having height zero (n>=0, 0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 2, 1, 9, 1, 28, 12, 1, 75, 120, 1, 186, 750, 120, 1, 441, 3780, 2100, 1, 1016, 16856, 21840, 1680, 1, 2295, 69552, 176400, 45360, 1, 5110, 272250, 1224720, 705600, 30240, 1, 11253, 1026300, 7692300, 8316000, 1164240, 1, 24564, 3762132, 45018600

Offset: 0

Alois P. Heinz, Nov 24 2007

			Triangle begins:
  1;
  1;
  1,     2;
  1,     9;
  1,    28,     12;
  1,    75,    120;
  1,   186,    750,     120;
  1,   441,   3780,    2100;
  1,  1016,  16856,   21840,   1680;
  1,  2295,  69552,  176400,  45360;
  1,  5110, 272250, 1224720, 705600, 30240;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
A. P. Heinz, Finding Two-Tree-Factor Elements of Tableau-Defined Monoids in Time O(n^3), Ed. S. G. Akl, F. Fiala, W. W. Koczkodaj: Advances in Computing and Information, ICCI90 Niagara Falls, LNCS 468, Springer-Verlag (1990), pp. 120-128.

Columns k=1,2 give: A058877, A133386.
Row sums give: A000248.
T(2n,n) = A001813(n), T(2n+1,n) = A002691(n).
Reading the table by diagonals gives triangle A198204. - Peter Bala, Jul 31 2012
Cf. A235596.

/* As triangle */ [[Binomial(n,k)*Factorial(k)*StirlingSecond(n-k+1,k+1): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 06 2019

T:= (n,k)-> binomial(n,k)*k!*Stirling2(n-k+1,k+1): for n from 0 to 10 do lprint(seq(T(n, k), k=0..floor(n/2))) od;

nn=12;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[ Series[Exp[y x (Exp[x]-1)] Exp[x],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Feb 09 2013 *)
t[n_, k_] := Binomial[n, k]*k!*StirlingS2[n-k+1, k+1]; Table[t[n, k], {n, 0, 12}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

T(n,k) = C(n,k) * k! * stirling2(n-k+1,k+1).
E.g.f.: exp(y*x*(exp(x)-1))*exp(x). - Geoffrey Critzer, Feb 09 2013
Sum_{k=1..floor(n/2)} T(n,k) = A235596(n+1). - Alois P. Heinz, Jun 21 2019

A275707 Number of partial functions f:{1,2,...,n}->{1,2,...,n} such that every element in the domain of definition of f is mapped to a fixed point or to an element that is undefined by f.

Original entry on oeis.org

1, 2, 8, 38, 216, 1402, 10156, 80838, 698704, 6498674, 64579284, 681642238, 7605025720, 89318058858, 1100376445564, 14176837311158, 190498308591264, 2663482511782114, 38667106019619748, 581765160424218606, 9055862445043643656, 145619330650420134362

Offset: 0

Geoffrey Critzer, Aug 06 2016

nonn

			G.f. = 1 + 2*x + 8*x^2 + 38*x^3 + 216*x^4 + 1402*x^5 + 10156*x^6 + ...
a(2) = 8 because there are 9 = A000169(3) partial functions on a set with 2 elements and all of them have the stated property except 1->2,2->1.

Alois P. Heinz, Table of n, a(n) for n = 0..524

Column 2 of A187105.

Cf. A000027, A000169, A000248, A351762, A355501.

a:= n-> add(binomial(n, k)*add(binomial(n-k, f)*
        (f+k)^(n-k-f), f=0..n-k), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 07 2016

nn = 20; Range[0, nn]! CoefficientList[Series[ Exp[z Exp[z]]^2, {z, 0, nn}], z]
Table[Sum[BellY[n, k, 2 Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

x='x+O('x^33); Vec(serlaplace(exp(2*x*exp(x)))) \\ Joerg Arndt, Nov 10 2016

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022

a(n) = sum(k=0, n, 2^k*k^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Jul 04 2022

E.g.f.: A(x)^2 = exp(2*B(x)) where A(x) is the e.g.f. for A000248 and B(x) is the e.g.f. for A000027.
E.g.f.: exp(2*x*exp(x)). - Joerg Arndt, Nov 10 2016
a(0) = 1; a(n) = Sum_{k=1..n} 2*k*binomial(n-1,k-1)*a(n-k). - Ilya Gutkovskiy, Nov 24 2017
From Seiichi Manyama, Jul 04 2022: (Start)
G.f.: Sum_{k>=0} (2 * x)^k/(1 - k*x)^(k+1).
a(n) = Sum_{k=0..n} 2^k * k^(n-k) * binomial(n,k). (End)
a(n) ~ n^(n + 1/2) * exp(2*r*exp(r) - r/2 - n) / (sqrt(2*(1 + 3*r + r^2)) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n)/2^(3/2)). - Vaclav Kotesovec, Jul 06 2022

A336227 a(0) = 1; a(n) = n * Sum_{k=0..n-1} binomial(n-1,k)^2 * a(k).

Original entry on oeis.org

1, 1, 4, 27, 292, 4425, 89106, 2280901, 71928872, 2728450017, 122145511510, 6354868381521, 379376236939404, 25710543779239501, 1960001963705060926, 166753195643254805565, 15724259680648667902096, 1633462474351643785483457, 185931510605274506452763166

Offset: 0

Ilya Gutkovskiy, Jul 12 2020

nonn

Cf. A000248, A023998.

a[0] = 1; a[n_] := a[n] = n Sum[Binomial[n - 1, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * [x^n] exp(sqrt(x) * BesselI(1,2*sqrt(x))).

A360782 Expansion of Sum_{k>=0} x^k / (1 - k*x^2)^(k+1).

Original entry on oeis.org

1, 1, 1, 3, 7, 16, 45, 125, 363, 1127, 3561, 11696, 39727, 138113, 494213, 1811075, 6784115, 25985928, 101520833, 404305549, 1640002039, 6767576175, 28395916893, 121048681024, 523902418555, 2300906314849, 10248029334297, 46266088140291

Offset: 0

Seiichi Manyama, Feb 20 2023

Cf. A000248, A360783.

Cf. A000045, A360592.

Join[{1}, Table[Sum[Binomial[n-k,k] * (n-2*k)^k, {k,0,n/2}], {n,1,30}]] (* Vaclav Kotesovec, Feb 21 2023 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k*x^2)^(k+1)))

a(n) = sum(k=0, n\2, (n-2*k)^k*binomial(n-k, k));

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n-k,k).

Previous Showing 31-40 of 105 results. Next

cp's OEIS Frontend

A135312 Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 2 for all x.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A209801 The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^2 in the equality sense.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A292978 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Ruby

Formula

A354550 Expansion of e.g.f. exp( x * exp(x^2/2) ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A354551 Expansion of e.g.f. exp( x * exp(x^3/6) ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

A133399 Triangle T(n,k)=number of forests of labeled rooted trees with n nodes, containing exactly k trees of height one, all others having height zero (n>=0, 0<=k<=floor(n/2)).

Views

Author

Keywords

Examples

Links