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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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

• T. D. Noe, Table of n, a(n) for n = 1..100
• Index entries for sequences related to card matching

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