A089460 -id:A089460 - cp's OEIS frontend

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

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

A089463 Triangle, read by rows, of coefficients for the third iteration of the hyperbinomial transform.

Original entry on oeis.org

1, 3, 1, 15, 6, 1, 108, 45, 9, 1, 1029, 432, 90, 12, 1, 12288, 5145, 1080, 150, 15, 1, 177147, 73728, 15435, 2160, 225, 18, 1, 3000000, 1240029, 258048, 36015, 3780, 315, 21, 1, 58461513, 24000000, 4960116, 688128, 72030, 6048, 420, 24, 1, 1289945088

Offset: 0

Paul D. Hanna, Nov 05 2003

Equals the matrix cube of A088956 when treated as a lower triangular matrix. The 3rd hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = Sum_{k=0..n} T(n,k)*b(k), where T(n,k) = 3*(n-k+3)^(n-k-1)*C(n,k). Given a table in which the n-th row is the n-th binomial transform of the first row, then the 3rd hyperbinomial transform of any diagonal results in the 3rd diagonal lower in the table.

			Rows begin:
  {1},
  {3,1},
  {15,6,1},
  {108,45,9,1},
  {1029,432,90,12,1},
  {12288,5145,1080,150,15,1},
  {177147,73728,15435,2160,225,18,1},
  {3000000,1240029,258048,36015,3780,315,21,1},..

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A089464(row sums), A089465(diagonal), A089460, A088956.

Flatten[Table[3(n-k+3)^(n-k-1) Binomial[n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 26 2013 *)

for(n=0,10, for(k=0,n, print1(3*(n-k+3)^(n-k-1)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 17 2017

T(n, k) = 3*(n-k+3)^(n-k-1)*C(n, k).
E.g.f.: exp(x*y)*(-LambertW(-y)/y)^3.
Note: (-LambertW(-y)/y)^3 = Sum_{n>=0} 3*(n+3)^(n-1)*y^n/n!.

A089462 2nd hyperbinomial transform of A001858.

Original entry on oeis.org

1, 3, 14, 93, 822, 9193, 125292, 2022555, 37829468, 805712859, 19270873704, 511742870653, 14946235170120, 476314240239633, 16451368229689808, 612254102183085627, 24428043107239133712, 1040281158638494489075

Offset: 0

Paul D. Hanna, Nov 05 2003

nonn

A001858 enumerates forests of labeled trees with n nodes and shifts 1 place left under the hyperbinomial transform.

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A001858, A089460.

Table[Sum[Sum[Binomial[m, j]*Binomial[n, n - m - j + 1]*(n + 2)^(n - m - j + 1)*(m + j)!/(-2)^j, {j, 0, m}]/m!, {m, 0, n + 1}], {n, 0, 50}] (* G. C. Greubel, Nov 18 2017 *)

a(n)=if(n<0,0,sum(m=0,n+1,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j+1)*(n+2)^(n-m-j+1)*(m+j)!/(-2)^j)/m!))

a(n) = Sum_{k=0..n} 2*(n-k+2)^(n-k-1)*C(n, k)*A001858(k).
a(n) = Sum_{m=0..(n+1)} ( Sum_{j=0..m} C(m, j)*C(n, n-m-j+1)*(n+2)^(n-m-j+1)*(m+j)!/(-2)^j)/m!.
a(n) ~ 2 * exp(5/2) * n^(n-1). - Vaclav Kotesovec, Oct 11 2020

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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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A089463 Triangle, read by rows, of coefficients for the third iteration of the hyperbinomial transform.

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A089462 2nd hyperbinomial transform of A001858.

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