A144303 -id:A144303 - cp's OEIS frontend

A088957 Hyperbinomial transform of the sequence of 1's.

1, 2, 6, 29, 212, 2117, 26830, 412015, 7433032, 154076201, 3608522954, 94238893883, 2715385121740, 85574061070045, 2928110179818478, 108110945014584623, 4284188833355367440, 181370804507130015569, 8169524599872649117330, 390114757072969964280163

Offset: 0

Paul D. Hanna, Oct 26 2003

nonn

See A088956 for the definition of the hyperbinomial transform.
a(n) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1.  The triangle A088956 classifies these functions according to the number of undefined elements in the domain.  The triangle A144289 classifies these functions according to the number of edges in their digraph representation (considering the empty function to have 1 edge).  The triangle A203092 classifies these functions according to the number of connected components. - Geoffrey Critzer, Dec 29 2011
a(n) is the number of rooted subtrees (for a fixed root) in the complete graph on n+1 vertices: a(3) = 29 is the number of rooted subtrees in K_4: 1 of size 1, 3 of size 2, 9 of size 3, and 16 spanning subtrees. - Alex Chin, Jul 25 2013 [corrected by Marko Riedel, Mar 31 2019]
From Gus Wiseman, Jan 28 2024: (Start)
Also the number of labeled loop-graphs on n vertices such that it is possible to choose a different vertex from each edge in exactly one way. For example, the a(3) = 29 uniquely choosable loop-graphs (loops shown as singletons) are:
  {}  {1}  {1,2}  {1,12}   {1,2,13}  {1,12,13}
      {2}  {1,3}  {1,13}   {1,2,23}  {1,12,23}
      {3}  {2,3}  {2,12}   {1,3,12}  {1,13,23}
                  {2,23}   {1,3,23}  {2,12,13}
                  {3,13}   {2,3,12}  {2,12,23}
                  {3,23}   {2,3,13}  {2,13,23}
                  {1,2,3}            {3,12,13}
                                     {3,12,23}
                                     {3,13,23}
(End)

			a(5) = 2117 = 1296 + 625 + 160 + 30 + 5 + 1 = sum of row 5 of triangle A088956.

Alois P. Heinz, Table of n, a(n) for n = 0..387
Marko Riedel et al., Proof of e.g.f. of sequence.

Cf. A088956 (triangle).
Row sums of A144289. - Alois P. Heinz, Jun 01 2009
Cf. A086331, A000169.
Column k=1 of A144303. - Alois P. Heinz, Oct 30 2012
The covering case is A000272, also the case of exactly n edges.
Without the choice condition we have A006125 (shifted left).
The unlabeled version is A087803.
The choosable version is A368927, covering A369140, loopless A133686.
The non-choosable version is A369141, covering A369142, loopless A367867.
Cf. A000081, A000085, A057500, A062740, A137916, A277473, A322661, A367904, A368596, A368597, A368924.

a088957 = sum . a088956_row  -- Reinhard Zumkeller, Jul 07 2013

a:= n-> add((n-j+1)^(n-j-1)*binomial(n,j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

nn = 16; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[Exp[x] Exp[t], {x, 0, nn}], x]  (* Geoffrey Critzer, Dec 29 2011 *)
With[{nmax = 50}, CoefficientList[Series[-LambertW[-x]*Exp[x]/x, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)

x='x+O('x^10); Vec(serlaplace(-lambertw(-x)*exp(x)/x)) \\ G. C. Greubel, Nov 14 2017

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k).
E.g.f.: A(x) = exp(x+sum(n>=1, n^(n-1)*x^n/n!)).
E.g.f.: -LambertW(-x)*exp(x)/x. - Vladeta Jovovic, Oct 27 2003
a(n) ~ exp(1+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013
Binomial transform of A000272. - Gus Wiseman, Jan 25 2024

A089461 Hyperbinomial transform of A088957. Also the row sums of triangle A089460, which lists the coefficients for the second hyperbinomial transform.

Original entry on oeis.org

1, 3, 13, 81, 689, 7553, 101961, 1639529, 30640257, 653150529, 15649353929, 416495026841, 12193949444193, 389572905351425, 13488730646528265, 503205102139969977, 20123584054543823105, 858863606297804378753, 38967500492977755457161, 1872974608860684814735385

Offset: 0

Paul D. Hanna, Nov 05 2003

nonn

a(n) is also the number of subtrees of the complete graph K_{n+1} which contain a fixed edge. For n=2, the a(2)=3 solutions are the 3 subtrees of complete graph K_3 which contain a fixed edge (i.e. the edge itself and 2 copies of K_{1,2}). - Kellie J. MacPhee, Jul 25 2013

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A088957, A089460 (triangle).

Column k=2 of A144303. - Alois P. Heinz, Oct 30 2012

a:= n-> add(2*(n-j+2)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

CoefficientList[Series[E^x*(-LambertW[-x]/x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)

x='x+O('x^50); Vec(serlaplace(exp(x)*(-lambertw(-x)/x)^2)) \\ G. C. Greubel, Nov 16 2017

a(n) = Sum_{k=0..n} 2*(n-k+2)^(n-k-1)*C(n, k).
E.g.f.: exp(x)*(-LambertW(-x)/x)^2.
a(n) ~ 2*exp(2+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013

A089464 Hyperbinomial transform of A089461. Also the row sums of triangle A089463, which lists the coefficients for the third hyperbinomial transform.

Original entry on oeis.org

1, 4, 22, 163, 1564, 18679, 268714, 4538209, 88188280, 1940666635, 47744244286, 1299383450941, 38777402351476, 1259552677645903, 44247546748659130, 1671904534990870369, 67624237153933934704, 2915628368081840175379, 133499617770334938670198

Offset: 0

Paul D. Hanna, Nov 05 2003

nonn

a(n) is also the number of subtrees of the complete graph K_{n+2} which contain 2 fixed adjacent edges (i.e. a fixed K_{1,2}). For n=2, the a(2)=4 solutions are the 4 subtrees of K_4 which contain 2 fixed adjacent edges (i.e. those 2 edges, 1 copy of K_{1,3}, and 2 copies of P_4). - Kellie J. MacPhee, Jul 25 2013

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A089461, A089463 (triangle).

Column k=3 of A144303.

a:= n-> add(3*(n-j+3)^(n-j-1)*binomial(n,j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

Table[Sum[3(n-k+3)^(n-k-1) Binomial[n,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Dec 04 2011 *)
CoefficientList[Series[E^x*(-LambertW[-x]/x)^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)

x='x+O('x^50); Vec(serlaplace(exp(x)*(-lambertw(-x)/x)^3)) \\ G. C. Greubel, Nov 16 2017

a(n) = Sum_{k=0..n} 3*(n-k+3)^(n-k-1)*C(n, k).
E.g.f.: exp(x)*(-LambertW(-x)/x)^3.
a(n) ~ 3*exp(3+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013

A218496 4th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 5, 33, 281, 2993, 38705, 592489, 10516441, 212841889, 4845154913, 122664558905, 3421333467689, 104297273041969, 3451364116327249, 123251578626936841, 4725537745859375705, 193647372258547916609, 8447809104669814884545, 390938955429073736493145

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=4 of A144303.

a:= n-> add(4*(n-j+4)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

E.g.f.: exp(x) * (-LambertW(-x)/x)^4.
a(n) = Sum_{j=0..n} 4 * (n-j+4)^(n-j-1) * C(n,j).
Hyperbinomial transform of A089464.
a(n) ~ 4*exp(4+exp(-1))*n^(n-1). - Vaclav Kotesovec, Aug 16 2013

A218497 5th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 6, 46, 441, 5156, 71801, 1166886, 21756251, 458803176, 10814534541, 282098765426, 8074875680471, 251807768368956, 8501320507058801, 309046115586282726, 12039399243732745851, 500492026353038459216, 22119195334250297991701, 1035767312348853244634586

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=5 of A144303.

a:= n-> add(5*(n-j+5)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[5*(n-j+5)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^5.
a(n) = Sum_{j=0..n} 5 * (n-j+5)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218496.
a(n) ~ 5*exp(5+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218498 6th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 7, 61, 649, 8257, 123217, 2120545, 41484625, 911339617, 22249542241, 598364232529, 17591851634353, 561695417002225, 19366094448215665, 717377453802538753, 28423991158962139873, 1199873992182732076225, 53772852099331738315969, 2550272224743737587911025

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=6 of A144303.

a:= n-> add(6*(n-j+6)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[6*(n-j+6)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^6.
a(n) = A(n,k) = Sum_{j=0..n} 6 * (n-j+6)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218497.
a(n) ~ 6*exp(6+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218499 7th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 8, 78, 911, 12524, 199403, 3624706, 74300269, 1699264792, 42964199279, 1191492782054, 35994106307321, 1177389200637028, 41482632276082915, 1566911405137366450, 63190697224460246477, 2710704012199936430000, 123277690401078017104343, 5925900498827152433216446

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=7 of A144303.

a:= n-> add(7*(n-j+7)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[7*(n-j+7)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^7.
a(n) = Sum_{j=0..n} 7 * (n-j+7)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218498.
a(n) ~ 7*exp(7+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218500 8th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 9, 97, 1233, 18209, 308129, 5901489, 126560849, 3010775745, 78805945665, 2253470828561, 69959985025841, 2345132738183841, 84468280694319713, 3254988169237833585, 133676275015986223569, 5830402582814375609729, 269227430712934320151169

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=8 of A144303.

a:= n-> add(8*(n-j+8)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[8*(n-j+8)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)
With[{nn=20},CoefficientList[Series[Exp[x](-LambertW[-x]/x)^8,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 04 2019 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^8.
a(n) = Sum_{j=0..n} 8 * (n-j+8)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218499.
a(n) ~ 8*exp(8+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218501 9th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 10, 118, 1621, 25588, 458605, 9232894, 206835751, 5113191304, 138473150833, 4081818946330, 130223467785619, 4473867764956204, 164772507070721989, 6479598382677480286, 271083794667222927655, 12026359894442420178064, 564099525344446492486105

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=9 of A144303.

a:= n-> add(9*(n-j+9)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[9*(n-j+9)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^9.
a(n) = Sum_{j=0..n} 9 * (n-j+9)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218500.
a(n) ~ 9*exp(9+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A218502 10th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 11, 141, 2081, 34961, 661601, 13970521, 326429401, 8377176001, 234573153281, 7125155956601, 233554674134441, 8223284332647361, 309711995280132001, 12430859603012736601, 529915231307371964201, 23918971999180778999681, 1139982481554110359552001

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=10 of A144303.

a:= n-> add(10*(n-j+10)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[10*(n-j+10)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^10.
a(n) = Sum_{j=0..n} 10 * (n-j+10)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218501.
a(n) ~ 10*exp(10+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

cp's OEIS Frontend

A088957 Hyperbinomial transform of the sequence of 1's.

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A089461 Hyperbinomial transform of A088957. Also the row sums of triangle A089460, which lists the coefficients for the second hyperbinomial transform.

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A089464 Hyperbinomial transform of A089461. Also the row sums of triangle A089463, which lists the coefficients for the third hyperbinomial transform.

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A218496 4th iteration of the hyperbinomial transform on the sequence of 1's.

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A218497 5th iteration of the hyperbinomial transform on the sequence of 1's.

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A218498 6th iteration of the hyperbinomial transform on the sequence of 1's.

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A218499 7th iteration of the hyperbinomial transform on the sequence of 1's.

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