A082395 Number of shifted Young tableaux with height <= 3.

1, 1, 2, 3, 6, 12, 27, 63, 154, 386, 989, 2574, 6787, 18085, 48622, 131719, 359194, 985186, 2715973, 7521568, 20915257, 58373587, 163462816, 459136810, 1293223231, 3651864607, 10336625732, 29321683083, 83344398534, 237344961292, 677087183363, 1934730432503

Offset: 1

Author

Fouad IBN MAJDOUB HASSANI, Apr 14 2003

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 27*x^7 + 63*x^8 + 154*x^9 + 386*x^10 + 989*x^11 + 2574*x^12 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• FouadIbn-Majdoub-Hassani, Combinatoire de polyominos et des tableaux décalés oscillants, Thèse de Doctorat, 1996, Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.

Crossrefs

Partial sums of A005043.

Cf. A363555.

Programs

Mathematica

Table[Sum[(-1)^(k+1)*Binomial[n, k]*Binomial[k-1, Floor[k/2]],{k,1,n}],{n,1,20}]
RecurrenceTable[{3*(n-2)*a[n-3]+(2-n)*a[n-2]+(4-3n)*a[n-1]+n*a[n]==0,a[1]==1,a[2]==1,a[3]==2},a,{n,20}] (* Vaclav Kotesovec, Oct 02 2012 *)

Sage

def A082395():
    a, b, s, n = 1, 0, 1, 1
    yield a
    while True:
        s += b
        yield s
        n += 1
        a, b = b, (2*b+3*a)*(n-1)/(n+1)
A082395_list = A082395()
[next(A082395_list) for i in range(30)] # Peter Luschny, Sep 24 2014

Formula

a(n) = Sum_{k=1..n}(-1)^(k+1)*binomial(n, k)*binomial(k-1, floor(k/2)). - Vladeta Jovovic, Sep 18 2003

Recurrence: 3*(n-2)*a(n-3)+(2-n)*a(n-2)+(4-3*n)*a(n-1)+n*a(n)=0. - Vaclav Kotesovec, Oct 02 2012

Asymptotic: a(n) ~ 3^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 02 2012

From Paul D. Hanna, Jul 08 2023: (Start)

G.f. A(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*(1-x^2)).

G.f. A(x) = A(x)^2 + (1 - A(x)^2)*x + (A(x) - A(x)^2)*x^2 + A(x)^2*x^3.

G.f. A(x) satisfies 0 = Sum_{n>=0} (-1)^n * x^(n*(n-3)/2) * A(x)^n / Product_{k=0..n+1} (1 - x^k*A(x)). (End)

Extensions

More terms from Vladeta Jovovic, Sep 18 2003