A089466 -id:A089466 - cp's OEIS frontend

A089467 Hyperbinomial transform of A089466 and also the inverse hyperbinomial transform of A089468.

1, 2, 8, 52, 478, 5706, 83824, 1461944, 29510268, 676549450, 17361810016, 492999348348, 15345359136232, 519525230896322, 19005788951346240, 747102849650454256, 31404054519248544016, 1405608808807797838866, 66741852193123060505728, 3350816586986433907218500, 177352811048578736727396576

Offset: 0

Paul D. Hanna, Nov 08 2003

nonn

See A088956 for the definition of the hyperbinomial transform.

Cf. A089466, A089468, A088956.

Flatten[{1, Table[Sum[Sum[Binomial[m, j] * Binomial[n, n-m-j] * n^(n-m-j) * (m+j)! / (-2)^j / m!, {j,0,m}], {m,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, Oct 11 2020 *)

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

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k)*A089466(k).
a(n) = Sum_{k=0..n} -(n-k-1)^(n-k-1)*C(n, k)*A089468(k).
a(n) = Sum_{m=0..n} (Sum_{j=0..m} C(m, j)*C(n, n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!.
a(n) ~ exp(1/2) * n^n. - Vaclav Kotesovec, Oct 11 2020

More terms from Michel Marcus, Jan 12 2025

A089468 Hyperbinomial transform of A089467 and also the 2nd hyperbinomial transform of A089466.

Original entry on oeis.org

1, 3, 15, 110, 1083, 13482, 203569, 3618540, 74058105, 1715620148, 44384718879, 1268498827752, 39692276983555, 1349678904881400, 49556966130059553, 1954156038072106448, 82363978221026323761, 3695194039210436996400

Offset: 0

Paul D. Hanna, Nov 08 2003

nonn

See A088956 for the definition of the hyperbinomial transform.

Cf. A089466, A089467, A088956.

CoefficientList[Series[(LambertW[-x]^2*E^(-1/2*LambertW[-x]^2))/(x^2*(1+LambertW[-x])), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)

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

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k)*A089467(k).
a(n) = Sum_{k=0..n} 2*(n-k+2)^(n-k-1)*C(n, k)*A089466(k).
a(n) = Sum_{m=0..n} (Sum_{j=0..m} C(m, j)*C(n, n-m-j)*(n+1)^(n-m-j)*(m+j)!/(-2)^j)/m!.
E.g.f.: (LambertW(-x)^2*exp(-1/2*LambertW(-x)^2))/(x^2*(1+LambertW(-x))). - Vladeta Jovovic, Oct 26 2004
a(n) ~ exp(3/2)*n^n. - Vaclav Kotesovec, Jul 09 2013

A071207 Triangular array T(n,k) read by rows, giving number of rooted trees on the vertex set {1..n+1} where k children of the root have a label smaller than the label of the root.

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 27, 27, 9, 1, 256, 256, 96, 16, 1, 3125, 3125, 1250, 250, 25, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 823543, 823543, 352947, 84035, 12005, 1029, 49, 1, 16777216, 16777216, 7340032, 1835008, 286720, 28672, 1792, 64, 1, 387420489

Offset: 0

Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

The n-th term of the n-th binomial transform of a sequence {b} is given by {d} where d(n) = sum(k=0,n,T(n,k)*b(k)) and T(n,k)=binomial(n,k)*n^(n-k); such diagonals are related to the hyperbinomial transform (A088956). - Paul D. Hanna, Nov 04 2003
T(n,k) gives the number of divisors of A181555(n) with (n-k) distinct prime factors. See also A001221, A146289, A146290, A181567. - Matthew Vandermast, Oct 31 2010
T(n,k) is the number of partial functions on {1,2,...,n} leaving exactly k elements undefined.  Row sums = A000169. - Geoffrey Critzer, Jan 08 2012
As a triangular matrix, transforms rows into diagonals in the table of coefficients of successive iterations of x/(1-x). - Paul D. Hanna, Jan 19 2014
Also the number of rooted trees on n+1 labeled vertices in which some specified vertex (say, vertex 1) has k children. - Alan Sokal, Jul 22 2022

			1
1     1
4     4     1
27    27    9     1
256   256   96    16    1
3125  3125  1250  250   25    1
46656 46656 19440 4320  540   36    1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
Alan D. Sokal, A remark on the enumeration of rooted labeled trees, arXiv:1910.14519 [math.CO], 2019 and Discrete Math. 343, 111865 (2020).

Cf. A000169, A000312, A089466, A088956, A166900.

T:= (n, k)-> binomial(n, k)*n^(n-k): seq(seq(T(n, k), k=0..n), n=0..10);

Prepend[Flatten[ Table[Table[Binomial[n, k] n^(n - k), {k, 0, n}], {n, 1, 8}]], 1]  (* Geoffrey Critzer, Jan 08 2012 *)

T(n,k)=if(k<0 || k>n,0,binomial(n,k)*n^(n-k))

/* Transforms rows into diagonals in the iterations of x/(1-x): */
{T(n, k)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x/(1-x+x*O(x^(m+2))))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x; for(i=1, r, F=subst(F, x, x/(1-x+x*O(x^(m+2))))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 19 2014

T(n,k) = binomial(n, k)*n^(n-k).

E.g.f.: (-LambertW(-y)/y)^x/(1+LambertW(-y)). - Vladeta Jovovic

Name edited by Alan Sokal, Jul 22 2022

A285270 a(n) = H_n(n), where H_n is the physicist's n-th Hermite polynomial.

Original entry on oeis.org

1, 2, 14, 180, 3340, 80600, 2389704, 83965616, 3409634960, 157077960480, 8093278209760, 461113571640128, 28784033772836544, 1953535902100115840, 143219579014652040320, 11279408109860685024000, 949705205977314865582336, 85131076752851318807814656, 8094279370190580822082014720

Offset: 0

Natan Arie Consigli, May 24 2017

nonn

			Knowing that H_3(x) = 8x^3-12x, a(3) = H_3(3) = 8*3^3-12*3 = 180.

Wikipedia, Hermite polynomial

Cf. A089466 (probabilist's variant).

Cf. A349066, A349067, A349068, A349069.

Table[HermiteH[n, n], {n, 0, 18}] (* Michael De Vlieger, May 25 2017 *)

a(n) = polhermite(n, n); \\ Michel Marcus, May 25 2017

from sympy import hermite
def a(n): return hermite(n, n) # Indranil Ghosh, May 25 2017

a(n) ~ exp(-1/4) * 2^n * n^n. - Vaclav Kotesovec, Nov 07 2021

More terms from Michel Marcus, May 25 2017

A185025 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 3, 1, 18, 9, 163, 90, 3, 1950, 1100, 75, 28821, 16245, 1575, 15, 505876, 283122, 33810, 735, 10270569, 5699932, 780150, 26460, 105, 236644092, 130267440, 19615932, 884520, 8505, 6098971555, 3332614725, 538325550, 29619450, 467775, 945

Offset: 0

Geoffrey Critzer, Dec 24 2012

It appears that as n gets large, row n conforms to a Poisson distribution with mean = 1/2. In other words, as n gets large, T(n,k) approaches n^n/(2^k*k!*e^(1/2)).

			Triangle begins:
           1;
           1;
           3,          1;
          18,          9;
         163,         90,         3;
        1950,       1100,        75;
       28821,      16245,      1575,       15;
      505876,     283122,     33810,      735;
    10270569,    5699932,    780150,    26460,    105;
   236644092,  130267440,  19615932,   884520,   8505;
  6098971555, 3332614725, 538325550, 29619450, 467775, 945;
  ...

Column k=0 gives A089466.

Cf. A000169, A081131.

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; Range[0,nn]! CoefficientList[Series[Exp[t^2/2(y-1)]/(1-t), {x,0,nn}], {x,y}]//Grid

E.g.f.: exp((T(x)^2/2)*(y-1))/(1 - T(x)) where T(x) is the e.g.f. for A000169.

Sum_{k=1..floor(n/2)} k * T(n,k) = A081131(n).

A334014 Array read by antidiagonals: T(n,k) is the number of functions f: X->Y, where X is a subset of Y, |X| = n, |Y| = n+k, such that for every x in X, f(f(x)) != x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 2, 1, 3, 8, 18, 30, 1, 4, 15, 52, 163, 444, 1, 5, 24, 110, 478, 1950, 7360, 1, 6, 35, 198, 1083, 5706, 28821, 138690, 1, 7, 48, 322, 2110, 13482, 83824, 505876, 2954364, 1, 8, 63, 488, 3715, 27768, 203569, 1461944, 10270569, 70469000, 1, 9, 80, 702, 6078, 51894, 436656, 3618540, 29510268, 236644092, 1864204416, 1, 10, 99, 970, 9403, 90150, 854485, 8003950, 74058105, 676549450, 6098971555, 54224221050

Offset: 0

Mason C. Hart, Apr 14 2020

Comes up in the study of the Zen Stare game (see description at A134362).

T(k,n-k)*binomial(n,k)*(n-k-1)!! is the number of different possible Zen Stare rounds with n starting players and k winners.

			Array begins:
=======================================================
n\k |    0     1     2      3      4      5       6
----+--------------------------------------------------
  0 |    1     1     1      1      1      1       1 ...
  1 |    0     1     2      3      4      5       6 ...
  2 |    0     3     8     15     24     35      48 ...
  3 |    2    18    52    110    198    322     488 ...
  4 |   30   163   478   1083   2110   3715    6078 ...
  5 |  444  1950  5706  13482  27768  51894   90150 ...
  6 | 7360 28821 83824 203569 436656 854485 1557376 ...
  ...
T(2,2) = 8; This because given X = {A,B}, Y = {A,B,C,D}. The only functions f: X->Y that meet the requirement are:
f(A) = C, f(B) = C
f(A) = D, f(B) = D
f(A) = D, f(B) = C
f(A) = C, f(B) = D
f(A) = B, f(B) = C
f(A) = B, f(B) = D
f(A) = C, f(B) = A
f(A) = D, f(B) = A

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=0..3 are A000012, A001477, A005563, A058794.
Columns k=0..4 are A134362, A089466, A089467, A089468, A220690(n+2).
Cf. A000169, A065440.

T(n,k)={my(w=-lambertw(-x + O(x^max(4,1+n)))); n!*polcoef(exp((k-1)*w - w^2/2)/(1-w), n)} \\ Andrew Howroyd, Apr 15 2020

T(n,k) = Sum_{i=0..n} k^(n-i)*binomial(n,i)*T(i,n-i); This means that with a constant n, T(n,k) is a polynomial of k.
T(n,0) = A134362(n).
T(0,k) = 1.
For odd n, Sum_{k=1..(n+1)/2} T(2*k-1,n-2*k+1)*binomial(n,2*k-1)*(n-2*k)!! = (n-1)^n.
E.g.f. of k-th column: exp((k-1)*W(x) - W(x)^2/2)/(1-W(x)) where W(x) is the e.g.f. of A000169. - Andrew Howroyd, Apr 15 2020

cp's OEIS Frontend

A089467 Hyperbinomial transform of A089466 and also the inverse hyperbinomial transform of A089468.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A089468 Hyperbinomial transform of A089467 and also the 2nd hyperbinomial transform of A089466.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A071207 Triangular array T(n,k) read by rows, giving number of rooted trees on the vertex set {1..n+1} where k children of the root have a label smaller than the label of the root.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A285270 a(n) = H_n(n), where H_n is the physicist's n-th Hermite polynomial.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A185025 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A334014 Array read by antidiagonals: T(n,k) is the number of functions f: X->Y, where X is a subset of Y, |X| = n, |Y| = n+k, such that for every x in X, f(f(x)) != x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula