A052512 -id:A052512 - cp's OEIS frontend

A000949 Number of forests with n nodes and height at most 2.

1, 1, 3, 16, 101, 756, 6607, 65794, 733833, 9046648, 121961051, 1782690174, 28055070397, 472594822324, 8479144213191, 161340195463066, 3243707386310033, 68679247688467056, 1526976223741111987, 35557878951515668726, 865217354118762606021

Offset: 0

N. J. A. Sloane

nonn

Equivalently, the number of mappings from a set of n elements into itself where f(f(x)) = f(f(f(x))). - Chad Brewbaker, Mar 26 2014

			G.f. = 1 + x + 3*x^2 + 16*x^3 + 101*x^4 + 756*x^5 + 6607*x^6 + 65794*x^7 + ... - _Michael Somos_, Jul 03 2018

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000950, A000951, A052512-A052514.

Column k=2 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
a[ n_] := If[ n < 0, 0, 1 + n! Sum[ Sum[ k^(n - m - k) m^k / (k! (n - m - k)!), {k, n - m}] / m!, {m, n - 1}]]; (* Michael Somos, Jul 03 2018 *)

a(n):=n!*sum(sum((k^(n-m-k)*m^k)/(k!*(n-m-k)!),k,1,n-m)/m!,m,1,n-1)+1; /* Vladimir Kruchinin, May 28 2011 */

x='x+O('x^66); Vec(serlaplace(exp(x*exp(x*exp(x))))) /* show terms with a(0)=1 */ /* Joerg Arndt, May 28 2011 */

E.g.f.: exp(x*exp(x*exp(x))).

a(n) = n!*sum(m=1..n-1, sum(k=1..n-m, (k^(n-m-k)*m^k)/(k!*(n-m-k)!))/m!)+1. - Vladimir Kruchinin, May 28 2011

More terms from Vladeta Jovovic, Apr 07 2001

A000951 Number of forests with n nodes and height at most 4.

Original entry on oeis.org

1, 3, 16, 125, 1296, 16087, 229384, 3687609, 66025360, 1303751051, 28151798544, 659841763957, 16681231615816, 452357366282655, 13095632549137576, 403040561722348913, 13138626717852194976, 452179922268565180819, 16381932383826669204640

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000949, A000950, A052512-A052514.

Column k=4 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x*Exp[x]]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)

E.g.f.: exp(x*exp(x*exp(x*exp(x*exp(x))))).

More terms from Vladeta Jovovic, Apr 07 2001

A185298 Expansion of e.g.f. xexp(x)exp(x*exp(x)).

Original entry on oeis.org

0, 1, 4, 18, 92, 520, 3222, 21700, 157544, 1224576, 10133450, 88843084, 821832156, 7992373168, 81458868974, 867700216380, 9636146477648, 111323478770560, 1335253363581330, 16598183219157772, 213488758730421380, 2837046652845555696, 38899888173340835894

Offset: 0

Geoffrey Critzer, Feb 20 2011

nonn

a(n) is the number of ways to designate an element in each block of the set partitions of {1,2,...,n} and then designate a block.
Inverse binomial transform: b(n) = Sum (-1)^(n-k)*C(n,k)*a(k), k=0..n of A052512. - Alexander R. Povolotsky, Oct 01 2011
Number of pointed set partitions of pointed sets k[1...k...n] for any point k. - Gus Wiseman, Sep 27 2015
Exponential series reversal gives A207833 with alternating signs: 1, -4, 30, -332, 4880, ... . - Vladimir Reshetnikov, Aug 04 2019

			The a(2) = 4 pointed set partitions are 1[1[12]], 1[1[1]2[2]], 2[1[1]2[2]], 2[2[12]].
The a(3) = 18 pointed set partitions are 1[1[123]], 1[1[1]2[23]], 1[1[1]3[23]], 1[1[12]3[3]], 1[1[13]2[2]], 1[1[1]2[2]3[3]], 2[2[123]], 2[1[1]2[23]], 2[1[13]2[2]], 2[2[2]3[13]], 2[2[12]3[3]], 2[1[1]2[2]3[3]], 3[3[123]], 3[1[1]3[23]], 3[1[12]3[3]], 3[2[2]3[13]], 3[2[12]3[3]], 3[1[1]2[2]3[3]].

G. C. Greubel, Table of n, a(n) for n = 0..535
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A052512, A080108, A207833, A262671.

nn=30; a=x Exp[x]; Range[0,nn]! CoefficientList[Series[a Exp[a], {x,0,nn}],x]

x='x+O('x^33); concat([0],Vec(serlaplace(x*exp(x)*exp(x*exp(x))))) \\ Joerg Arndt, Oct 04 2015

E.g.f.: A(A(x)) where A(x) = x*exp(x).
a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-k+1). - Vladimir Kruchinin, Sep 23 2011
O.g.f.: Sum_{k>=1} k*x^k/(1 - k*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) ~ exp(r*exp(r) + r - n) * n^(n + 1/2) / (r^(n - 1/2) * sqrt(1 + exp(r)*(1 + 3*r + r^2))), where r = 2*LambertW(exp(1/4)*sqrt(n)/2) - 1/2 + 1/(16*LambertW(exp(1/4)*sqrt(n)/2)^2 + LambertW(exp(1/4)*sqrt(n)/2) - 1). - Vaclav Kotesovec, Mar 21 2023

A000950 Number of forests with n nodes and height at most 3.

Original entry on oeis.org

1, 3, 16, 125, 1176, 12847, 160504, 2261289, 35464816, 612419291, 11539360944, 235469524237, 5170808565976, 121535533284999, 3043254281853496, 80852247370051793, 2270951670959226336, 67221368736302224819, 2091039845329887687136

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000949-A000951, A052512-A052514.

Column k=3 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)

E.g.f.: exp(x*exp(x*exp(x*exp(x)))).

More terms from Vladeta Jovovic, Apr 07 2001

A116071 Triangle T, read by rows, equal to Pascal's triangle to the matrix power of Pascal's triangle, so that T = C^C, where C(n,k) = binomial(n,k) and T(n,k) = A000248(n-k)*C(n,k).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 41, 40, 18, 4, 1, 196, 205, 100, 30, 5, 1, 1057, 1176, 615, 200, 45, 6, 1, 6322, 7399, 4116, 1435, 350, 63, 7, 1, 41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1, 293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1

Offset: 0

Paul D. Hanna, Feb 03 2006

Column 0 = A000248 (Number of forests with n nodes and height at most 1).
Column 1 = A052512 (Number of labeled trees of height 2).
Row sums = A080108 (Sum_{k=1..n} k^(n-k) * C(n-1,k-1)).
Central terms = A116072(n) = (n+1) * A000108(n) * A000248(n).
From Peter Bala, Sep 13 2012: (Start)
For commuting lower unitriangular matrix A and lower triangular matrix B we define A raised to the matrix power B, denoted by A^B, to be the lower unitriangular matrix Exp(B*Log(A)). Here Exp denotes the matrix exponential defined by the power series
  Exp(A) = 1 + A + A^2/2! + A^3/3! + ...
and the matrix logarithm Log(A) is defined by the series
  Log(A) = (A-1) - 1/2*(A-1)^2/2 + 1/3*(A-1)^3 - ....
Let A = [f(x),x] and B = [g(x),x] be exponential Riordan arrays in the Appell subgroup and suppose f(0) = 1. Then A and B commute and A^B is the exponential Riordan array [exp(g(x)*log(f(x))),x], also belonging to the Appell group. In the present case we are taking A = B = [exp(x),x], equal to the Pascal triangle A007318.
For any lower unitriangular matrix A (with, say, rational entries) the infinite tower of powers A^(A^(A^(...))) is well-defined (and also has rational entries). An example is given in the Formula section. (End)

			E.g.f.: E(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2/2!
  + (10 + 9*y + 3*y^2 + y^3)*x^3/3!
  + (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4/4!
  + (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5/5! +...
where E(x,y) = exp(x*y) * exp(x*exp(x)).
O.g.f.: A(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2
  + (10 + 9*y + 3*y^2 + y^3)*x^3
  + (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4
  + (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5 +...
where
A(x,y) = 1/(1-x*y) + x/(1-x*(y+1))^2 + x^2/(1-x*(y+2))^3 + x^3/(1-x*(y+3))^4 + x^4/(1-x*(y+4))^5 + x^5/(1-x*(y+5))^6 + x^6/(1-x*(y+6))^7 + x^7/(1-x*(y+7))^8 +...
Triangle begins:
  1;
  1, 1;
  3, 2, 1;
  10, 9, 3, 1;
  41, 40, 18, 4, 1;
  196, 205, 100, 30, 5, 1;
  1057, 1176, 615, 200, 45, 6, 1;
  6322, 7399, 4116, 1435, 350, 63, 7, 1;
  41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1;
  293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080 (rows 0..45 of flattened triangle).

Cf. A000248, A052512, A080108, A116072, A215652.

Cf. A080108, A216689, A240165, A245834, A245835, A216973.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[# Exp[#]]&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

/* By definition C^C: */
{T(n,k)=local(A, C=matrix(n+1,n+1,r,c,binomial(r-1,c-1)), L=matrix(n+1,n+1,r,c,if(r==c+1,c))); A=sum(m=0,n,L^m*C^m/m!); A[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From e.g.f.: */
{T(n,k)=local(A=1);A=exp( x*y + x*exp(x +x*O(x^n)) );n!*polcoeff(polcoeff(A, n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From o.g.f. (Paul D. Hanna, Aug 03 2014): */
{T(n,k)=local(A=1);A=sum(k=0, n, x^k/(1 - x*(k+y) +x*O(x^n))^(k+1));polcoeff(polcoeff(A, n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From row polynomials (Paul D. Hanna, Aug 03 2014): */
{T(n,k)=local(R);R=sum(k=0,n,(k+y)^(n-k)*binomial(n,k));polcoeff(R,k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From formula for T(n,k) (Paul D. Hanna, Aug 03 2014): */
{T(n,k) = sum(j=0,n-k, binomial(n,j) * binomial(n-j,k) * j^(n-k-j))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

E.g.f.: exp( x*exp(x) + x*y ).
From Peter Bala, Sep 13 2012: (Start)
Exponential Riordan array [exp(x*exp(x)),x] belonging to the Appell group. Thus the e.g.f. for the k-th column of the triangle is x^k/k!*exp(x*exp(x)).
The inverse array, denote it by X, is a signed version of A215652. The infinite tower of matrix powers X^(X^(X^(...))) equals the inverse of Pascal's triangle. (End)
O.g.f.: Sum_{n>=0} x^n / (1 - x*(n+y))^(n+1). - Paul D. Hanna, Aug 03 2014
G.f. for row n: Sum_{k=0..n} binomial(n,k) * (k + y)^(n-k) for n>=0. - Paul D. Hanna, Aug 03 2014
T(n,k) = Sum_{j=0..n-k} C(n,j) * C(n-j,k) * j^(n-k-j) = A000248(n-k)*C(n,k). - Paul D. Hanna, Aug 03 2014
Infinitesimal generator is A216973. - Peter Bala, Feb 13 2017

A362788 Triangle read by rows, T(n, k) = RisingFactorial(n - k, k) * Stirling2(n - k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 3, 6, 0, 4, 36, 0, 5, 140, 60, 0, 6, 450, 720, 0, 7, 1302, 5250, 840, 0, 8, 3528, 30240, 16800, 0, 9, 9144, 151704, 196560, 15120, 0, 10, 22950, 695520, 1764000, 453600, 0, 11, 56210, 2994750, 13471920, 7761600, 332640

Offset: 0

Peter Luschny, May 04 2023

			Triangle T(n, k) starts:
[0] 1;
[1] 0;
[2] 0, 1;
[3] 0, 2;
[4] 0, 3,    6;
[5] 0, 4,   36;
[6] 0, 5,  140,    60;
[7] 0, 6,  450,   720;
[8] 0, 7, 1302,  5250,   840;
[9] 0, 8, 3528, 30240, 16800;

Cf. A052512 (row sums), A362369, A362789.

T := (n, k) -> pochhammer(n - k, k) * Stirling2(n - k, k):
seq(seq(T(n, k), k = 0..iquo(n,2)), n = 0..12);

def A362788(n, k):
    return rising_factorial(n - k, k) * stirling_number2(n - k, k)
for n in range(10):
    print([A362788(n, k) for k in range(n//2 + 1)])

A216255 Triangle read by rows: T(n,k) is the number of labeled rooted trees of height at most 2 that have exactly k nodes at a distance 2 from the root; n>=1, 0<=k<=n-1.

Original entry on oeis.org

1, 2, 0, 3, 6, 0, 4, 24, 12, 0, 5, 60, 120, 20, 0, 6, 120, 540, 480, 30, 0, 7, 210, 1680, 3780, 1680, 42, 0, 8, 336, 4200, 17920, 22680, 5376, 56, 0, 9, 504, 9072, 63000, 161280, 122472, 16128, 72, 0, 10, 720, 17640, 181440, 787500, 1290240, 612360, 46080, 90, 0

Offset: 1

Geoffrey Critzer, Mar 15 2013

Row sums = A052512.

Column k=1: A007531.

			1;
2,  0;
3,  6,   0;
4,  24,  12,    0;
5,  60,  120,   20,     0;
6,  120, 540,   480,    30,     0;
7,  210, 1680,  3780,   1680,   42,      0;
8,  336, 4200,  17920,  22680,  5376,    56,     0;
9,  504, 9072,  63000,  161280, 122472,  16128,  72,    0;
10, 720, 17640, 181440, 787500, 1290240, 612360, 46080, 90, 0;
T(4,1) = 24 because there is only one unlabeled tree on 4 nodes with exactly 1 node at distance two from the root.  It has 24 labelings.
.......o......
....../.\.....
.....o...o....
..../.........
...o..........

Alois P. Heinz, Rows n = 1..141, flattened

T:= (n, k)-> n*binomial(n-1,k)*(n-k-1)^k:
seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, Mar 15 2013

nn=10;a=NestList[x Exp[#]&,y x,nn];f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[a[[3]],{x,0,nn}],{x,y}]]//Grid

E.g.f.: x*exp(x*exp(y*x)).

T(n,k) = n*C(n-1,k)*(n-k-1)^k. - Alois P. Heinz, Mar 15 2013

A220233 Triangular array read by rows. T(n,k) is the number of labeled rooted trees of height at most 2 with exactly k leaves at a distance 1 from the root, n>=1, 0<=k<=n-1.

Original entry on oeis.org

0, 0, 2, 6, 0, 3, 12, 24, 0, 4, 80, 60, 60, 0, 5, 390, 480, 180, 120, 0, 6, 2352, 2730, 1680, 420, 210, 0, 7, 15176, 18816, 10920, 4480, 840, 336, 0, 8, 106416, 136584, 84672, 32760, 10080, 1512, 504, 0, 9, 801450, 1064160, 682920, 282240, 81900, 20160, 2520, 720, 0, 10

Offset: 1

Geoffrey Critzer, Dec 08 2012

Row sums = A052512 for n>1. Column for k=0 is A220232.

			Triangle T(n,k) begins:
     0;
     0,    2;
     6,    0,    3;
    12,   24,    0,   4;
    80,   60,   60,   0,   5;
   390,  480,  180, 120,   0, 6;
  2352, 2730, 1680, 420, 210, 0, 7;
  ...

nn=7;f[list_]:=Select[list,#>0&];a=x (Exp[x]-1+y);Prepend[Drop[Map[Insert[#,0,-2]&,Map[f,Range[0,nn]!CoefficientList[Series[x (Exp[a]-1+y),{x,0,nn}],{x,y}]]],2],{0}]//Grid

E.g.f.: x*(exp(x*(exp(x) -1 + y)) - 1 + y) (letting T(1,1)=1).

A259760 Triangle read by rows: T(n,k) is the number of partial idempotent mappings (of an n-chain) with breadth exactly k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 9, 10, 1, 4, 18, 40, 41, 1, 5, 30, 100, 205, 196, 1, 6, 45, 200, 615, 1176, 1057, 1, 7, 63, 350, 1435, 4116, 7399, 6322, 1, 8, 84, 560, 2870, 10976, 29596, 50576, 41393, 1, 9, 108, 840, 5166, 24696, 88788, 227592, 372537, 293608

Offset: 0

Wafa AlNadabi, Jul 04 2015

			T(3,2) = 9 because there are exactly 9 partial idempotent mappings (of a 3-chain) with breadth exactly 2, namely: (12-->11), (12-->22), (12-->12), (13-->11), (13-->33), (13-->13), (23-->22), (23-->33), (23-->23).
Triangle starts:
1;
1, 1;
1, 2, 3;
1, 3, 9, 10;
1, 4, 18, 40, 41;
...

F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).

Haoliang Wang, Robert Simon, The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.

Cf. A000248, A052512.

Row sums give A080108(n+1).

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n,k)*sum(m=0, k, binomial(k,m)*m^(k-m)), ", ");); print(););} \\ Michel Marcus, Jul 15 2015

T(n,k) = binomial(n,k) * Sum_{m=0..k} binomial(k,m)*m^(k-m).

More terms from Michel Marcus, Jul 15 2015

cp's OEIS Frontend

A000949 Number of forests with n nodes and height at most 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

Extensions

A000951 Number of forests with n nodes and height at most 4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185298 Expansion of e.g.f. x*exp(x)*exp(x*exp(x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A000950 Number of forests with n nodes and height at most 3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A116071 Triangle T, read by rows, equal to Pascal's triangle to the matrix power of Pascal's triangle, so that T = C^C, where C(n,k) = binomial(n,k) and T(n,k) = A000248(n-k)*C(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

Formula

A362788 Triangle read by rows, T(n, k) = RisingFactorial(n - k, k) * Stirling2(n - k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

SageMath

A216255 Triangle read by rows: T(n,k) is the number of labeled rooted trees of height at most 2 that have exactly k nodes at a distance 2 from the root; n>=1, 0<=k<=n-1.

Views

Author

Keywords

Comments

A185298 Expansion of e.g.f. xexp(x)exp(x*exp(x)).