A003161 - OEIS

%I M1931 #44 Apr 06 2023 08:33:54

%S 1,1,2,9,36,190,980,5705,33040,204876,1268568,8209278,53105976,

%T 354331692,2364239592,16140234825,110206067400,765868074400,

%U 5323547715200,37525317999884,264576141331216,1886768082651816,13458185494436592,96906387191038334,697931136204820336

%N A binomial coefficient sum.

%C The number of triples of standard tableaux of the same shape of height less than or equal to 2. - _Mike Zabrocki_, Mar 29 2007

%C From _Peter Bala_, Mar 20 2023: (Start)

%C For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(3,n)}. For other cases see A361887 ({S(5,n)}) and A361890 ({S(7,n)}).

%C Gould (1974) proposed the problem of showing that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).

%C Conjecture: Let b(n) = a(2*n-1). Then the supercongruence b(n*p^k) == b(n*p^(k-1)) (mod p^(3*k)) holds for positive integers n and k and all primes p >= 5. (End)

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

%H Alois P. Heinz, <a href="/A003161/b003161.txt">Table of n, a(n) for n = 0..1116</a>

%H F. Bergeron, L. Favreau and D. Krob, <a href="http://dx.doi.org/10.1016/0012-365X(94)00148-C">Conjectures on the enumeration of tableaux of bounded height</a>, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.

%H H. W. Gould, <a href="http://www.jstor.org/stable/2976965">Problem E2384</a>, Amer. Math. Monthly, 81 (1974), 170-171.

%F a(n) = Sum_{k=0..n} A120730(n,k)^3. - _Philippe Deléham_, Oct 18 2008

%F G.f.: hypergeometric expression with an anti-derivative, see Maple program. - _Mark van Hoeij_, May 06 2013

%F Recurrence: n*(n+1)^3*(7*n^2 - 14*n + 3)*a(n) = - n*(7*n^5 - 112*n^4 + 206*n^3 + 8*n^2 - 125*n + 48)*a(n-1) + 16*(n-1)*(28*n^5 - 133*n^4 + 194*n^3 - 33*n^2 - 120*n + 61)*a(n-2) + 64*(n-2)^3*(n-1)*(7*n^2 - 4)*a(n-3). - _Vaclav Kotesovec_, Mar 06 2014

%F a(n) ~ 2^(3*n+9/2) / (9 * Pi^(3/2) * n^(5/2)). - _Vaclav Kotesovec_, Mar 06 2014

%F a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^3. - _Alois P. Heinz_, Oct 17 2022

%p ogf := ((8*x-1)*(8*x+1)*hypergeom([1/4, 1/4],[1],64*x^2)^2/(x+1)-3*Int((16*x-5)*hypergeom([1/4, 1/4],[1],64*x^2)^2/(x+1)^2,x)+1)/(16*x);

%p series(ogf,x=0,30); # _Mark van Hoeij_, May 06 2013

%t Table[Sum[(Binomial[n, k]-Binomial[n, k-1])^3,{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 06 2014 *)

%o (PARI) a(n)=sum(k=0,n\2, (binomial(n,k)-binomial(n,k-1))^3) /* _Michael Somos_, Jun 02 2005 */

%Y Cf. A003162, A008315.

%Y Cf. A001405, A000108, A129123, A183069.

%Y Column k=3 of A357824.

%Y Cf. A361887, A361890.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_