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 1-4 of 4 results.

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

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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

A000535 Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).

Original entry on oeis.org

0, 27, 378, 4536, 48600, 489780, 4738104, 44535456, 409752432, 3708359550, 33125746500, 292779558720, 2565087894720, 22307854940280, 192788833482000, 1657111548654720, 14176605442521312, 120779466450505758, 1025230099571720676, 8674221270307971600
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 two (2) fixed points. E.g., if AAAAABBBBBCCCCC n=3*5 letters permutations then two fixed points n5=48600. - 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. A000279, A000489.
Cf. A033581.

Programs

  • Mathematica
    a[n_] := 3*Binomial[n, 2]*Sum[Binomial[n, k+2]*Binomial[n, k]*Binomial[n-2, k], {k, 0, n-2}] + 3n^2*Sum[Binomial[n, k+1]*Binomial[n-1, k+1]*Binomial[ n-1, k], {k, 0, n-2}] (* Jean-François Alcover, Feb 09 2016 *)
  • PARI
    A000535(n)=3*binomial(n,2)*sum(k=0,n-2,binomial(n,k+2)*binomial(n,k)*binomial(n-2,k))+3*n^2*sum(k=0,n-2,binomial(n,k+1)*binomial(n-1,k+1)*binomial(n-1,k)) \\ M. F. Hasler, Sep 30 2015

Formula

a(n) = 3*binomial(n, 2)*Sum_{k=0..n-2} binomial(n, k+2)*binomial(n, k)*binomial(n-2, k) + 3*n^2*Sum_{k=0..n-2} binomial(n, k+1)*binomial(n-1, k+1)*binomial(n-1, k).
a(n) = 3(n-1)*n^3 3F2(1-n, 1-n, 2-n; 2, 2; -1) + (3/4)(n-1)^2 n^2 3F2(2-n, 2-n, -n; 1, 3; -1), where 3F2 is the hypergeometric function 3F2. - Jean-François Alcover, Feb 09 2016
a(n) ~ 3^(3/2) * 2^(3*n - 2) * n / Pi. - Vaclav Kotesovec, Jun 10 2019

Extensions

More terms from Vladeta Jovovic, Apr 26 2000
More terms from Emeric Deutsch, Feb 19 2004
More explicit definition by M. F. Hasler, Sep 22 2015

A104473 a(n) = binomial(n+2,2)*binomial(n+6,2).

Original entry on oeis.org

15, 63, 168, 360, 675, 1155, 1848, 2808, 4095, 5775, 7920, 10608, 13923, 17955, 22800, 28560, 35343, 43263, 52440, 63000, 75075, 88803, 104328, 121800, 141375, 163215, 187488, 214368, 244035, 276675, 312480, 351648, 394383, 440895, 491400, 546120, 605283, 669123
Offset: 0

Views

Author

Zerinvary Lajos, Apr 18 2005

Keywords

Examples

			a(0) = C(0+2,2)*C(0+6,2) = C(2,2)*C(6,2) = 1*15 = 155.
a(6) = 1*3*5 + 2*4*6 + 3*5*7 + 4*6*8 + 5*7*9 + 6*8*10 + 7*9*11 = 1848.

Links

Crossrefs

Cf. A000217, A000579, A033275, A034856, A062264.
Subsequence of A085780.

Programs

  • Magma
    [Binomial(n+2, 2)*Binomial(n+6, 2): n in [0..50]]; // Vincenzo Librandi, Apr 28 2014
  • Mathematica
    f[n_] := Binomial[n + 2, 2] Binomial[n + 6, 2]; Table[f[n], {n,0,40}] (* Robert G. Wilson v, Apr 20 2005 *)
CoefficientList[Series[3 (5-4*x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Apr 28 2014 *)
  • PARI
    a(n)=binomial(n+2,2)*binomial(n+6,2) \\ Charles R Greathouse IV, Jun 07 2013
  • SageMath
    def A104473(n): return binomial(n+2,2)*binomial(n+6,2)
print([A104473(n) for n in range(51)]) # G. C. Greubel, Mar 05 2025

Formula

a(n) = (1/4)*(n+1)*(n+2)*(n+5)*(n+6).
a(n) = A034856(n+2)^2 - 1. - J. M. Bergot, Dec 14 2010
G.f.: 3*(5-4*x+x^2)/(1-x)^5. - Colin Barker, Sep 21 2012
a(n) = Sum_{i=1..n+1} i*(i+2)*(i+4). - Bruno Berselli, Apr 28 2014
a(n) = A000217(n)*A000217(n+4) = 3*A033275(n+4). - R. J. Mathar, Nov 29 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 43/450.
Sum_{n>=0} (-1)^n/a(n) = 16*log(2)/15 - 154/225. (End)
From G. C. Greubel, Mar 05 2025: (Start)
a(n) = 90*A000579(n+6)/A000279(n+3).
E.g.f.: (1/4)*(60 + 192*x + 114*x^2 + 20*x^3 + x^4)*exp(x). (End)

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 1-4 of 4 results.

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

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