%I #16 Jul 18 2023 11:22:03

%S 0,0,0,1,4,16,64,236,818,2736,8934,28622,90324,281792,871556,2677750,

%T 8184383,24913238,75593383,228793147,691094857,2084237036,6277871658,

%U 18890568921,56798001639,170665733660,512554832309,1538718547049

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

%H T. Mansour and M. Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Shattuck/shattuck3.html">Counting Peaks and Valleys in a Partition of a Set</a>, J. Int. Seq. 13 (2010), 10.6.8, Table 1.

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,130,-280,471,-643,734,-701,575,-400,237,-121,49,-18,4,-1).

%F 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

%F 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

%p with(orthopoly) :

%p nmax := 15:

%p tpr := 1+x^2/2 :

%p k := 4:

%p g := x^k ;

%p for j from 1 to k do

%p if j> 1 then

%p g := g*( U(j-1,tpr)-(1+x)*U(j-2,tpr)) / ((1-x)*U(j-1,tpr)-U(j-2,tpr)) ;

%p else

%p # note that U(-1,.)=0, U(0,.)=1

%p g := g* U(j-1,tpr) / ((1-x)*U(j-1,tpr)) ;

%p end if;

%p end do:

%p simplify(%) ;

%p taylor(g,x=0,nmax+1) ;

%p gfun[seriestolist](%) ; # _R. J. Mathar_, Mar 11 2021

%Y Cf. A289692 (2 blocks), A289693 (3 blocks).

%K nonn,easy

%O 1,5

%A _R. J. Mathar_, Jul 09 2017