A003161 A binomial coefficient sum. (Formerly M1931)

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

0,3

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.

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

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 *)

Prog

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

Crossrefs

Cf. A003162, A008315.

Cf. A001405, A000108, A129123, A183069.

Column k=3 of A357824.

Cf. A361887, A361890.

Sequence in context: A137955 A150981 A150982 * A305206 A101610 A111601

Adjacent sequences: A003158 A003159 A003160 * A003162 A003163 A003164

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved