A056857 -id:A056857 - cp's OEIS frontend

A129324 Third column of PE^2.

0, 0, 1, 6, 36, 220, 1410, 9534, 68040, 511704, 4046310, 33560010, 291244668, 2638581972, 24901833866, 244333004790, 2487900487440, 26245651191600, 286408960814862, 3228529392965250, 37544229610105220, 449858650676764140

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A129324 := proc(n) A078937(n+1,2) ; end: seq(A129324(n),n=0..23) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078937[n + 1, 2];
a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,3]; with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,3].

E.g.f.: (x^2/2) * exp(2 * (exp(x) - 1)). - Ilya Gutkovskiy, Jul 11 2020

More terms from R. J. Mathar, May 30 2008

A129325 Fourth column of PE^2.

Original entry on oeis.org

0, 0, 0, 1, 8, 60, 440, 3290, 25424, 204120, 1705680, 14836470, 134240040, 1262060228, 12313382536, 124509169330, 1303109358880, 14098102762160, 157473907149600, 1813923418494126, 21523529286435000, 262809607270736540

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A129325 := proc(n) A078937(n+1,3) ; end: seq(A129325(n),n=0..27) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078937[n + 1, 3];
a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

m=matpascal(30)-matid(31); pe=matid(31)+sum(i=1,30,m^i/i!); A=pe^2; A[,4] \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,4] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,4]

More terms from R. J. Mathar and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

A129327 Second column of PE^3.

Original entry on oeis.org

0, 1, 6, 36, 228, 1545, 11196, 86457, 708504, 6136830, 55976430, 535904259, 5369146272, 56145107577, 611336534802, 6916529431620, 81152874393168, 985767316792449, 12376996566040980, 160399065135692073

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129327 := proc(n) A078938(n+1,1) ; end: seq(A129327(n),n=0..27) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 3], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,2 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,2]

More terms from R. J. Mathar, May 30 2008

A129328 Third column of PE^3.

Original entry on oeis.org

0, 0, 1, 9, 72, 570, 4635, 39186, 345828, 3188268, 30684150, 307870365, 3215425554, 34899450768, 393015753039, 4585024011015, 55332235452960, 689799432341928, 8871905851132041, 117581467377389310, 1603990651356920730

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129328 := proc(n) A078938(n+1,2) ; end: seq(A129328(n),n=0..27) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078938[n + 1, 2];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,3 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,3]

More terms from R. J. Mathar, May 30 2008

A129329 Fourth column of PE^3.

Original entry on oeis.org

0, 0, 0, 1, 12, 120, 1140, 10815, 104496, 1037484, 10627560, 112508550, 1231481460, 13933510734, 162864103584, 1965078765195, 24453461392080, 313549334233440, 4138796594051568, 56188737057169593, 783876449182595400

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129329 := proc(n) A078938(n+1,3) ; end: seq(A129329(n),n=0..27) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078938[n + 1, 3];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,4 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,4]

E.g.f.: (x^3/6) * exp(3 * (exp(x) - 1)). - Ilya Gutkovskiy, Jul 11 2020

More terms from R. J. Mathar, May 30 2008

A129331 Second column of PE^4.

Original entry on oeis.org

0, 1, 8, 60, 464, 3780, 32568, 296492, 2845088, 28695060, 303334920, 3351877628, 38622668400, 463036981732, 5764038605528, 74365952622540, 992720923710272, 13690497077256628, 194777994524434344, 2855149354656290716

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129331 := proc(n) A078939(n+1,1) ; end: seq(A129331(n),n=0..25) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 4], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,2 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,2]

More terms from R. J. Mathar, May 30 2008

A129332 Third column of PE^4.

Original entry on oeis.org

0, 0, 1, 12, 120, 1160, 11340, 113988, 1185968, 12802896, 143475300, 1668342060, 20111265768, 251047344600, 3241258872124, 43230289541460, 594927620980320, 8438127851537312, 123214473695309652, 1850390947982126268

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129332 := proc(n) A078939(n+1,2) ; end: seq(A129332(n),n=0..25) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
A078939[n_, c_] := Sum[A078938[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078939[n + 1, 2];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,3 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,3]

More terms from R. J. Mathar, May 30 2008

A129333 Fourth column of PE^4.

Original entry on oeis.org

0, 0, 0, 1, 16, 200, 2320, 26460, 303968, 3557904, 42676320, 526076100, 6673368240, 87148818328, 1171554274800, 16206294360620, 230561544221120, 3371256518888480, 50628767109223872, 780358333403627796

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129333 := proc(n) A078939(n+1,3) ; end: seq(A129333(n),n=0..25) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
A078939[n_, c_] := Sum[A078938[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078939[n + 1, 3];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,4 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,4]

More terms from R. J. Mathar, May 30 2008

A283424 Number T(n,k) of blocks of size >= k in all set partitions of [n], assuming that every set partition contains one block of size zero; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 15, 10, 4, 1, 52, 37, 17, 5, 1, 203, 151, 76, 26, 6, 1, 877, 674, 362, 137, 37, 7, 1, 4140, 3263, 1842, 750, 225, 50, 8, 1, 21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1, 115975, 94828, 57568, 25996, 8944, 2392, 502, 82, 10, 1

Offset: 0

Alois P. Heinz, May 14 2017

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			T(3,2) = 4 because the number of blocks of size >= 2 in all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+1+1+1+0 = 4.
Triangle T(n,k) begins:
      1;
      2,     1;
      5,     3,    1;
     15,    10,    4,    1;
     52,    37,   17,    5,    1;
    203,   151,   76,   26,    6,   1;
    877,   674,  362,  137,   37,   7,  1;
   4140,  3263, 1842,  750,  225,  50,  8, 1;
  21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000110(n+1), A138378 or A005493(n-1), A124325, A288785, A288786, A288787, A288788, A288789, A288790, A288791, A288792.
Row sums give A124427(n+1).
T(2n,n) gives A286896.
Cf. A005493, A056857, A056860, A070071, A224271, A285595.

T:= proc(n, k) option remember; `if`(k>n, 0,
      binomial(n, k)*combinat[bell](n-k)+T(n, k+1))
    end:
seq(seq(T(n, k), k=0..n), n=0..14);

T[n_, k_] := Sum[Binomial[n, j]*BellB[j], {j, 0, n - k}];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2018 *)

T(n,k) = Sum_{j=0..n-k} binomial(n,j) * Bell(j).
T(n,k) = Bell(n+1) - Sum_{j=0..k-1} binomial(n,j) * Bell(n-j).
T(n,k) = Sum_{j=k..n} A056857(n+1,j) = Sum_{j=k..n} A056860(n+1,n+1-j).
Sum_{k=0..n} T(n,k) = n*Bell(n)+Bell(n+1) = A124427(n+1).
Sum_{k=1..n} T(n,k) = n*Bell(n) = A070071(n).
T(n,0)-T(n,1) = Bell(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A224271(n+1). - Alois P. Heinz, May 17 2023

A139383 Number of n-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 2, 12, 154, 3455, 120196, 5995892, 406005804, 35839643175, 3998289746065, 550054365477936, 91478394767427823, 18091315306315315610, 4196205472500769304318, 1128136777063831105273242, 347994813261017613045578964, 122080313159891715442898099217

Offset: 0

Paul D. Hanna, Apr 16 2008

nonn

Define the matrix function matexps(M) to be exp(M)/exp(1). Then the number of k-level labeled rooted trees with n leaves is also column 0 of the triangle resulting from the n-th iteration of matexps on the Pascal matrix P, A007318. The resulting triangle is also S^n*P*S^-n, where S is the Stirling2 matrix A048993. This function can be coded in PARI as sum(k=0,200,1./k!*M^k)/exp(1), using exp(M) does not work. See A056857, which equals (1/e)*exp(P) or S*P*S^-1. - Gerald McGarvey, Aug 19 2009

			If we form a table from the family of sequences defined by:
number of k-level labeled rooted trees with n leaves,
then this sequence equals the diagonal in that table:
n=1:A000012=[1,1,1,1,1,1,1,1,1,1,...];
n=2:A000110=[1,2,5,15,52,203,877,4140,21147,115975,...];
n=3:A000258=[1,3,12,60,358,2471,19302,167894,1606137,...];
n=4:A000307=[1,4,22,154,1304,12915,146115,1855570,26097835,...];
n=5:A000357=[1,5,35,315,3455,44590,660665,11035095,204904830,...];
n=6:A000405=[1,6,51,561,7556,120196,2201856,45592666,1051951026,...];
n=7:A001669=[1,7,70,910,14532,274778,5995892,148154860,4085619622,...];
n=8:A081624=[1,8,92,1380,25488,558426,14140722,406005804,13024655442,...];
n=9:A081629=[1,9,117,1989,41709,1038975,29947185,979687005,35839643175,..].
Row n in the above table equals column 0 of matrix power A008277^n where A008277 = triangle of Stirling numbers of 2nd kind:
1;
1,1;
1,3,1;
1,7,6,1;
1,15,25,10,1;
1,31,90,65,15,1; ...
The name of this sequence is a generalization of the definition given in the above sequences by _Christian G. Bower_.

Alois P. Heinz, Table of n, a(n) for n = 0..247

Cf. A008277; A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629, A261280, A290354.

A diagonal of A144150.

A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
      add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
    end:
a:= n-> A(n, n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 14 2015
# second Maple program:
g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1+(g@@n)(x), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 31 2017
# third Maple program:
b:= proc(n, t, m) option remember; `if`(t=0, `if`(n<2, 1, 0),
     `if`(n=0, b(m, t-1, 0), m*b(n-1, t, m)+b(n-1, t, m+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

t[n_,m_]:=t[n,m] = If[m==1,1,Sum[StirlingS2[n,k]*t[k,m-1],{k,1,n}]]; Table[t[n,n],{n,1,20}] (* Vaclav Kotesovec, Aug 14 2015 after Vladimir Kruchinin *)

T(n,m):=if m=1 then 1 else sum(stirling2(n,i)*T(i,m-1),i,1,n);
makelist(T(n,n),n,1,7); /* Vladimir Kruchinin, May 19 2012 */

{a(n)=local(E=exp(x+x*O(x^n))-1,F=x); for(i=1,n,F=subst(F,x,E));n!*polcoeff(F,n)}

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def A(n, k): return 1 if n==0 or k==0 else sum(binomial(n - 1, j - 1)*A(j, k - 1)*A(n - j, k) for j in range(1, n + 1))
def a(n): return A(n, n - 1)
print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 07 2017, after Maple code

a(n) = T(n,n), T(n,m) = Sum_{i=1..n} Stirling2(n,i)*T(i,m-1), m>1, T(n,1)=1. - Vladimir Kruchinin, May 19 2012
a(n) = n! * [x^n] 1 + g^n(x), where g(x) = exp(x)-1. - Alois P. Heinz, Aug 14 2015
From Vaclav Kotesovec, Aug 14 2015: (Start)
Conjecture: a(n) ~ c * n^(2*n-5/6) / (2^(n-1) * exp(n)), where c = 2.86539...
a(n) ~ exp(-1) * A261280(n).
(End)

a(0)=1 prepended by Alois P. Heinz, Jul 31 2017

cp's OEIS Frontend

A129324 Third column of PE^2.

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A129325 Fourth column of PE^2.

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PARI

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A129327 Second column of PE^3.

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A129328 Third column of PE^3.

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A129329 Fourth column of PE^3.

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A129331 Second column of PE^4.

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A129332 Third column of PE^4.

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A129333 Fourth column of PE^4.