cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A089466 Inverse hyperbinomial transform of A089467.

Original entry on oeis.org

1, 1, 3, 18, 163, 1950, 28821, 505876, 10270569, 236644092, 6098971555, 173823708696, 5427760272507, 184267682837992, 6757353631762293, 266191329601854000, 11210291102456374801, 502602430218071545104, 23900770928782913595651, 1201581698963550283673632
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2003

Keywords

Comments

See A088956 for the definition of the hyperbinomial transform.
a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the functional digraph contains no cycles of length 2. - Geoffrey Critzer, Mar 21 2012

Links

Crossrefs

Cf. A089467, A088956.

Programs

  • Mathematica
    nn=20; t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; a=Log[1/(1-t)]; Range[0,nn]! CoefficientList[Series[Exp[a-t^2/2], {x,0,nn}], x] (* Geoffrey Critzer, Mar 21 2012 *)
  • PARI
    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!))
  • PARI
    a(n) = n! * sum(k=0, n\2, (-1/2)^k * n^(n - 2*k) / (k! * (n - 2*k)!)); \\ Daniel Suteu, Jun 19 2018

Formula

A089467(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k)*a(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.: exp(-(T(x))^2/2)/(1-T(x)), where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 21 2012
a(n) ~ exp(-1/2) * n^n. - Vaclav Kotesovec, Oct 08 2013
a(n) = n! * Sum_{k=0..floor(n/2)} (-1/2)^k * n^(n - 2*k) / (k! * (n - 2*k)!). - Daniel Suteu, Jun 19 2018

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

Views

Author

Paul D. Hanna, Nov 08 2003

Keywords

Comments

See A088956 for the definition of the hyperbinomial transform.

Crossrefs

Cf. A089466, A089467, A088956.

Programs

  • Mathematica
    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 *)
  • PARI
    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!))

Formula

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

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

Views

Author

Mason C. Hart, Apr 14 2020

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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
Showing 1-3 of 3 results.

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