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

1,4

Links

Table of n, a(n) for n=1..30.

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).

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)

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

Crossrefs

Cf. A289692, A289694.

Sequence in context: A193703 A289658 A180238 * A269684 A330079 A361423

Adjacent sequences: A289690 A289691 A289692 * A289694 A289695 A289696

Keyword

nonn

Author

R. J. Mathar, Jul 09 2017

Status

approved