A089461 -id:A089461 - cp's OEIS frontend

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

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

A144303 Square array A(n,m), n>=0, m>=0, read by antidiagonals: A(n,m) = n-th number of the m-th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 29, 1, 1, 5, 22, 81, 212, 1, 1, 6, 33, 163, 689, 2117, 1, 1, 7, 46, 281, 1564, 7553, 26830, 1, 1, 8, 61, 441, 2993, 18679, 101961, 412015, 1, 1, 9, 78, 649, 5156, 38705, 268714, 1639529, 7433032, 1, 1, 10, 97, 911, 8257, 71801, 592489, 4538209, 30640257, 154076201, 1

Offset: 0

Alois P. Heinz, Sep 17 2008, revised Oct 30 2012

See A088956 for the definition of the hyperbinomial transform.

A(n,m), n>=0, m>=0, is the number of subtrees of the complete graph K_{n+m} on n+m vertices containing a given, fixed subtree on m vertices. - Alex Chin, Jul 25 2013

			Square array begins:
  1,     1,      1,      1,      1,       1,       1, ...
  1,     2,      3,      4,      5,       6,       7, ...
  1,     6,     13,     22,     33,      46,      61, ...
  1,    29,     81,    163,    281,     441,     649, ...
  1,   212,    689,   1564,   2993,    5156,    8257, ...
  1,  2117,   7553,  18679,  38705,   71801,  123217, ...
  1, 26830, 101961, 268714, 592489, 1166886, 2120545, ...

Alois P. Heinz, Rows n = 0..140, flattened
N. J. A. Sloane, Transforms

Columns m=0-10 give: A000012, A088957, A089461, A089464, A218496, A218497, A218498, A218499, A218500, A218501, A218502.
Rows n=0-2 give: A000012, A000027, A028872.
Main diagonal gives A252766.
Cf. A007318, A088956.

hymtr:= proc(p) proc(n,m) `if`(m=0, p(n), m*add(
           p(k)*binomial(n, k) *(n-k+m)^(n-k-1), k=0..n))
        end end:
A:= hymtr(1):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[, 0] = 1; a[n, k_] := Sum[k*(n - j + k)^(n - j - 1)*Binomial[n, j], {j, 0, n}]; Table[a[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jun 24 2013 *)

E.g.f. of column k: exp(x) * (-LambertW(-x)/x)^k.

A(n,k) = Sum_{j=0..n} k * (n-j+k)^(n-j-1) * C(n,j).

A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.

Original entry on oeis.org

1, 2, 1, 8, 4, 1, 50, 24, 6, 1, 432, 200, 48, 8, 1, 4802, 2160, 500, 80, 10, 1, 65536, 28812, 6480, 1000, 120, 12, 1, 1062882, 458752, 100842, 15120, 1750, 168, 14, 1, 20000000, 8503056, 1835008, 268912, 30240, 2800, 224, 16, 1, 428717762, 180000000, 38263752, 5505024, 605052, 54432, 4200, 288, 18, 1

Offset: 0

Paul D. Hanna, Nov 05 2003

Equals the matrix square of A088956 when treated as a lower triangular matrix. The 2nd 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) = 2*(n-k+2)^(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 2nd hyperbinomial transform of any diagonal results in the diagonal located 2 diagonals lower in the table.

			Rows begin:
  {1},
  {2,1},
  {8,4,1},
  {50,24,6,1},
  {432,200,48,8,1},
  {4802,2160,500,80,10,1},
  {65536,28812,6480,1000,120,12,1},
  {1062882,458752,100842,15120,1750,168,14,1},..

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

Cf. A089461(row sums), A089462(diagonal), A089463, A088956.

Join[{1}, Table[Binomial[n, k]*2*(n - k + 2)^(n - k - 1), {n, 1, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 18 2017 *)

for(n=0,10, for(k=0,n, print1(2*(n-k+2)^(n-k-1)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 18 2017

T(n, k) = 2*(n-k+2)^(n-k-1)*C(n, k).
E.g.f.: exp(x*y)*(-LambertW(-y)/y)^2.
Note: (-LambertW(-y)/y)^2 = Sum_{n>=0} 2*(n+2)^(n-1)*y^n/n!.

A362522 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 3, 7, 49, 201, 2491, 14743, 266337, 2055889, 49051891, 466650471, 13873711633, 156839920537, 5591748678699, 73222243463671, 3046762637864641, 45346835284775073, 2158148557098011107, 35980450963558606279, 1928292118820446611441

Offset: 0

Seiichi Manyama, Apr 23 2023

Seiichi Manyama, Table of n, a(n) for n = 0..416
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A362523.

Cf. A089461, A362347.

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

E.g.f.: exp(x - LambertW(-x^2)) = -LambertW(-x^2)/x^2 * exp(x).

a(n) ~ sqrt(2) * (exp(2*exp(-1/2)) + (-1)^n) * n^(n-1) / exp(n/2 + exp(-1/2) - 1). - Vaclav Kotesovec, Aug 05 2025

cp's OEIS Frontend

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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A144303 Square array A(n,m), n>=0, m>=0, read by antidiagonals: A(n,m) = n-th number of the m-th iteration of the hyperbinomial transform on the sequence of 1's.

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A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.

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A362522 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (k! * (n-2*k)!).

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