A000489 - OEIS

%I M5025 N2168 #47 Nov 15 2023 08:04:59

%S 1,16,435,7136,99350,1234032,14219212,155251840,1628202762,

%T 16550991200,164111079110,1594594348800,15235525651840,

%U 143518352447680,1335670583147400,12301278983461376,112264111607438906,1016361486936571680,9136254276320346046

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

%C The definition uses notations of Riordan (1958), except for use of n instead of p. - _M. F. Hasler_, Sep 22 2015

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000489/b000489.txt">Table of n, a(n) for n = 1..100</a>

%H <a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>

%F 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).

%F 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

%F a(n) ~ 3*sqrt(3)*n^2*8^(n-1)/Pi. - _Vaclav Kotesovec_, Aug 07 2013

%F 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

%t 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 *)

%o (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

%o (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

%Y Cf. A000279, A000535.

%Y Cf. A059056 - A059071.

%Y Cf. A001181.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Apr 26 2000

%E More terms from _Emeric Deutsch_, Feb 19 2004

%E Definition made more precise by _M. F. Hasler_, Sep 22 2015