A289693 The number of partitions of [n] with exactly 3 blocks without peaks.

0, 0, 1, 3, 9, 27, 75, 197, 503, 1263, 3132, 7695, 18784, 45649, 110585, 267276, 644907, 1554208, 3742321, 9005265, 21659603, 52078400, 125186565, 300870586, 723010749, 1737273406, 4174084259, 10028409724, 24092769583, 57880137331

Offset: 1

Author

R. J. Mathar, Jul 09 2017

Keywords

• nonn

Links

• T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Table 1.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,24,-29,25,-17,9,-3,1).

Crossrefs

Cf. A289692, A289694.

Programs

Maple

with(orthopoly) :
nmax := 10:
tpr := 1+x^2/2 :
k := 3:
g := x^k ;
for j from 1 to k do
        if j> 1 then
                g := g*( U(j-1,tpr)-(1+x)*U(j-2,tpr)) / ((1-x)*U(j-1,tpr)-U(j-2,tpr)) ;
        else
                # note that U(-1,.)=0, U(0,.)=1
                g := g* U(j-1,tpr) / ((1-x)*U(j-1,tpr)) ;
        end if;
end do:
simplify(%) ;
taylor(g,x=0,nmax+1) ;
gfun[seriestolist](%) ; # R. J. Mathar, Mar 11 2021

Formula

From Colin Barker, Nov 07 2017: (Start)

G.f.: x^3*(1 - x + x^2)*(1 - 2*x + 3*x^2 - x^3 + x^4) / ((1 - x)*(1 - 2*x + x^2 - x^3)*(1 - 3*x + 3*x^2 - 4*x^3 + x^4 - x^5)).

a(n) = 6*a(n-1) - 15*a(n-2) + 24*a(n-3) - 29*a(n-4) + 25*a(n-5) - 17*a(n-6) + 9*a(n-7) - 3*a(n-8) + a(n-9) for n>9.

(End)