A289694 The number of partitions of [n] with exactly 4 blocks without peaks.

0, 0, 0, 1, 4, 16, 64, 236, 818, 2736, 8934, 28622, 90324, 281792, 871556, 2677750, 8184383, 24913238, 75593383, 228793147, 691094857, 2084237036, 6277871658, 18890568921, 56798001639, 170665733660, 512554832309, 1538718547049

Offset

1,5

Links

Table of n, a(n) for n=1..28.

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 (10,-45,130,-280,471,-643,734,-701,575,-400,237,-121,49,-18,4,-1).

Formula

G.f. x^4*(x^2-x+1)*(x^4-x^3+3*x^2-2*x+1)*(x^6-x^5+5*x^4-4*x^3+6*x^2-3*x+1) / ( (x-1)*(x^5-x^4+4*x^3-3*x^2+3*x-1)*(x^7-x^6+6*x^5-5*x^4+10*x^3-6*x^2+4*x-1)*(x^3-x^2+2*x-1) ). - R. J. Mathar, Mar 11 2021

a(n)= 10*a(n-1) -45*a(n-2) +130*a(n-3) -280*a(n-4) +471*a(n-5) -643*a(n-6) +734*a(n-7) -701*a(n-8) +575*a(n-9) -400*a(n-10) +237*a(n-11) -121*a(n-12) +49*a(n-13) -18*a(n-14) +4*a(n-15) -a(n-16). - R. J. Mathar, Mar 11 2021

Maple

with(orthopoly) :
nmax := 15:
tpr := 1+x^2/2 :
k := 4:
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 (2 blocks), A289693 (3 blocks).

Sequence in context: A248088 A294037 A228735 * A232425 A180239 A006811

Adjacent sequences: A289691 A289692 A289693 * A289695 A289696 A289697

Keyword

nonn,easy

Author

R. J. Mathar, Jul 09 2017

Status

approved