A379836 Number of pairs of adjacent equal parts in all complete compositions of n.

0, 0, 1, 2, 5, 12, 23, 54, 118, 258, 550, 1178, 2540, 5394, 11473, 24174, 51021, 107210, 225099, 471322, 985202, 2055542, 4281847, 8906676, 18500425, 38379246, 79516158, 164561560, 340179441, 702506576, 1449311429, 2987297778, 6151964642, 12658841766, 26027603925

Offset

0,4

Comments

An integer composition is complete if its set of parts covers an initial interval.

Links

Table of n, a(n) for n=0..34.

Formula

G.f.: B(x) = d/dz Sum_{k>0} C({1..k},x,z)|_{z=1} where C({s},x,z) = Sum_{i in {s}} ( C({s}-{i},x,z)*(x^i)/(1-(x^i)*(z-1)) )/(1 - Sum_{i in {s}} (x^i)/(1-(x^i)*(z-1))) with C({},x,z) = 1.

Example

The complete compositions of n = 4 are: (1,1,2), (1,2,1), (2,1,1), and (1,1,1,1); having a total of 5 pairs of equal adjacent parts giving a(4) = 5.

Prog

(PARI)
C_xz(s, N) = {my(x='x+O('x^N), g=if(#s <1, 1, sum(i=1, #s, C_xz(s[^i], N+1) * x^(s[i])/(1-(x^(s[i]))*(z-1)) )/(1-sum(i=1, #s, x^(s[i])/(1-(x^(s[i]))*(z-1)))))); return(g)}
B_xz(N) = {my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, h += C_xz(vector(j, i, i), N+1); j+=1); h}
P_xz(N) = Pol(B_xz(N), {x})
B_x(N) = {my(cx = deriv(P_xz(N), z), z=1); Vecrev(eval(cx))}
B_x(20)

Crossrefs

Cf. A011782, A106356, A107428, A107429, A373306, A374147, A374726, A377823.

Sequence in context: A096376 A357288 A326510 * A112287 A127787 A116733

Adjacent sequences: A379833 A379834 A379835 * A379837 A379838 A379839

Keyword

nonn

Author

John Tyler Rascoe, Jan 14 2025

Status

approved