A003161 A binomial coefficient sum.

1, 1, 2, 9, 36, 190, 980, 5705, 33040, 204876, 1268568, 8209278, 53105976, 354331692, 2364239592, 16140234825, 110206067400, 765868074400, 5323547715200, 37525317999884, 264576141331216, 1886768082651816, 13458185494436592, 96906387191038334, 697931136204820336

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The number of triples of standard tableaux of the same shape of height less than or equal to 2. - Mike Zabrocki, Mar 29 2007

From Peter Bala, Mar 20 2023: (Start)

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)}).

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).

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)

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1116
• F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
• H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

Crossrefs

Cf. A003162, A008315.

Cf. A001405, A000108, A129123, A183069.

Column k=3 of A357824.

Cf. A361887, A361890.

Programs

Maple

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);
series(ogf,x=0,30); # Mark van Hoeij, May 06 2013

Mathematica

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

PARI

a(n)=sum(k=0,n\2, (binomial(n,k)-binomial(n,k-1))^3) /* Michael Somos, Jun 02 2005 */

Formula

a(n) = Sum_{k=0..n} A120730(n,k)^3. - Philippe Deléham, Oct 18 2008

G.f.: hypergeometric expression with an anti-derivative, see Maple program. - Mark van Hoeij, May 06 2013

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

a(n) ~ 2^(3*n+9/2) / (9 * Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Mar 06 2014

a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^3. - Alois P. Heinz, Oct 17 2022