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.

A000279 Card matching: coefficients B[n,1] of t in the reduced hit polynomial A[n,n,n](t).

Original entry on oeis.org

3, 24, 216, 1824, 15150, 124416, 1014888, 8241792, 66724398, 538990800, 4346692680, 35009591040, 281699380560, 2264868936960, 18198009147600, 146142982814208, 1173123636533454, 9413509300965936, 75513633110271264, 605598295606296000, 4855626127979443908, 38924245740546950784
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Number of permutations of 3 distinct letters (ABC) each with n copies such that one (1) fixed points. E.g., if AAAAABBBBBCCCCC n=3*5 letters permutations then one fixed points n5=15150. - Zerinvary Lajos, Feb 02 2006
The definition uses notations of Riordan (1958), except for use of n instead of p. - M. F. Hasler, Sep 22 2015

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000489, A000535.
Cf. A033581.

Programs

  • Mathematica
    f[n_] := HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]; a[n_] := n^2*(f[n]+4*f[n-1])/(n+1); Array[a, 20] (* Jean-François Alcover, Mar 11 2014, after Mark van Hoeij *)
  • PARI
    A000279(n)=3*n*sum(k=0,n-1,binomial(n,k+1)*binomial(n,k)*binomial(n-1,k)) \\ M. F. Hasler, Sep 21 2015

Formula

a(n) = 3n * sum(C(n, k+1)*C(n, k)*C(n-1, k), k=0..n-1).
G.f.: x * (6*hypergeom([4/3, 5/3],[2],27*x^2/(1-2*x)^3)/(1-2*x)^3 - 3*hypergeom([2/3, 4/3],[1],27*x^2/(1-2*x)^3)/(1-2*x)^2). - Mark van Hoeij, Oct 23 2011
a(n) = n^2*(A000172(n)+4*A000172(n-1))/(n+1). - Mark van Hoeij, Oct 26 2011
a(n) ~ 8^n*sqrt(3)/Pi = 8^n*0.5513... - M. F. Hasler, Sep 21 2015
a(n) = 3n*A262407(n). - M. F. Hasler, Sep 23 2015

Extensions

More terms from Vladeta Jovovic, Apr 26 2000
More terms from Emeric Deutsch, Feb 19 2004
Three lines of data completed and more explicit definition by M. F. Hasler, Sep 21 2015

A000489 Card matching: Coefficients B[n,3] of t^3 in the reduced hit polynomial A[n,n,n](t).

Original entry on oeis.org

1, 16, 435, 7136, 99350, 1234032, 14219212, 155251840, 1628202762, 16550991200, 164111079110, 1594594348800, 15235525651840, 143518352447680, 1335670583147400, 12301278983461376, 112264111607438906, 1016361486936571680, 9136254276320346046
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The definition uses notations of Riordan (1958), except for use of n instead of p. - M. F. Hasler, Sep 22 2015

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000279, A000535.
Cf. A059056 - A059071.
Cf. A001181.

Programs

  • Magma
    [1, 16] cat [&+[3*Binomial(n, 3)*Binomial(n, k+3)*Binomial(n, k)*Binomial(n-3, k) + 6*n*Binomial(n, 2)*Binomial(n, k+1)*Binomial(n-1, k+2)*Binomial(n-2, k): k in [0..n-3]] + &+[n^3*Binomial(n-1, k)^3: k in [0..n-1]]: n in [3..20]]; // Vincenzo Librandi, Sep 22 2015
  • Mathematica
    a[n_] := 3*Binomial[n, 3]*Sum[Binomial[n, k + 3]*Binomial[n, k]*Binomial[n - 3, k], {k, 0, n - 3}] + 6 n*Binomial[n, 2]*Sum[Binomial[n, k + 1]*Binomial[n - 1, k + 2]*Binomial[n - 2, k], {k, 0, n - 3}] + n^3*Sum[Binomial[n - 1, k]^3, {k, 0, n - 1}]; Table[a[n], {n, 20}] (* T. D. Noe, Jun 20 2012 *)
  • PARI
    A000489(n)={3*binomial(n, 3)*sum(k=0,n-3,binomial(n, k+3)*binomial(n, k)*binomial(n-3, k))+6*n*binomial(n, 2)*sum(k=0,n-3,binomial(n, k+1)*binomial(n-1, k+2)*binomial(n-2, k))+n^3*sum(k=0,n-1,binomial(n-1, k)^3)} \\ M. F. Hasler, Sep 20 2015

Formula

a(n) = 3*binomial(n, 3)*sum(binomial(n, k+3)*binomial(n, k)*binomial(n-3, k), k=0..n-3) + 6n*binomial(n, 2)*sum(binomial(n, k+1)*binomial(n-1, k+2)*binomial(n-2, k), k=0..n-3) + n^3*sum(binomial(n-1, k)^3, k=0..n-1).
Recurrence: (n+3)*(243*n^7 - 1701*n^6 + 4239*n^5 - 4671*n^4 + 6042*n^3 - 17352*n^2 + 25032*n - 12016)*(n-1)^2*a(n) = n*(1701*n^9 - 6804*n^8 + 270*n^7 + 19116*n^6 + 35085*n^5 - 203640*n^4 + 324384*n^3 - 246736*n^2 + 75440*n - 5440)*a(n-1) + 8*n*(243*n^7 - 864*n^5 - 486*n^4 + 4233*n^3 - 5274*n^2 + 2460*n - 184)*(n-1)^2*a(n-2). - Vaclav Kotesovec, Aug 07 2013
a(n) ~ 3*sqrt(3)*n^2*8^(n-1)/Pi. - Vaclav Kotesovec, Aug 07 2013
a(n) = n^2*((27*n^3+54*n^2-57*n+8)*(n+2)*A001181(n)-(189*n^3+189*n^2-30*n+16)*(n-1)*A001181(n-1))/96. - Mark van Hoeij, Nov 14 2023

Extensions

More terms from Vladeta Jovovic, Apr 26 2000
More terms from Emeric Deutsch, Feb 19 2004
Definition made more precise by M. F. Hasler, Sep 22 2015

A262407 a(n) = Sum_{k=0..n-1} C(n,k+1)*C(n,k)*C(n-1,k).

Original entry on oeis.org

0, 1, 4, 24, 152, 1010, 6912, 48328, 343408, 2471274, 17966360, 131717960, 972488640, 7223061040, 53925450880, 404400203280, 3044645475296, 23002424245754, 174324246314184, 1324800580881952, 10093304926771600, 77073430602848316, 589761299099196224
Offset: 0

Views

Author

M. F. Hasler, Sep 21 2015

Keywords

Crossrefs

Cf. A000489, A000535, A000279.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n,
       ((21*n^3-49*n^2+30*n-8)*a(n-1)+
        (8*(n-1))*(n-2)*(3*n-1)*a(n-2))/
        ((3*n-4)*(n+1)*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2015
  • Mathematica
    f[n_]:=HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]; a[n_]:=n^2 (f[n] + 4 f[n - 1])/(3 n^2 + 3 n); Array[a, 25] (* Vincenzo Librandi, Sep 22 2015 *)
Table[Sum[Binomial[n,k+1]Binomial[n,k]Binomial[n-1,k],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Apr 09 2021 *)
  • PARI
    a(n)=sum(k=0, n-1, binomial(n, k+1)*binomial(n, k)*binomial(n-1, k))

Formula

a(n) = A000279(n)/(3*n) = (A000172(n)+4*A000172(n-1))*n/(3*(n+1)).
a(n) ~ 8^n/(sqrt(3)*Pi*n) as n -> oo.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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