A089468 -id:A089468 - 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

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.

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