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A003161 A binomial coefficient sum.
(Formerly M1931)
5
1, 1, 2, 9, 36, 190, 980, 5705, 33040, 204876, 1268568, 8209278, 53105976, 354331692, 2364239592, 16140234825, 110206067400, 765868074400, 5323547715200, 37525317999884, 264576141331216, 1886768082651816 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..21.

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=k<=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

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.

Cf. A001405, A000108, A129123.

Sequence in context: A137955 A150981 A150982 * A101610 A111601 A280351

Adjacent sequences:  A003158 A003159 A003160 * A003162 A003163 A003164

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 24 05:48 EST 2018. Contains 299597 sequences. (Running on oeis4.)