A000489 Card matching: Coefficients B[n,3] of t^3 in the reduced hit polynomial A[n,n,n](t). (Formerly M5025 N2168)

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

Offset

1,2

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

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000279, A000535.

Cf. A059056 - A059071.

Cf. A001181.

Sequence in context: A111921 A118994 A223686 * A075852 A260853 A068792

Adjacent sequences: A000486 A000487 A000488 * A000490 A000491 A000492

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved