A238872 Number of strongly unimodal compositions of n with absolute difference of successive parts = 1.

1, 1, 1, 3, 2, 3, 3, 4, 3, 6, 4, 3, 5, 6, 4, 9, 5, 3, 7, 7, 5, 9, 6, 6, 8, 9, 5, 9, 8, 6, 10, 6, 5, 15, 8, 9, 10, 7, 7, 12, 10, 3, 11, 15, 7, 15, 8, 6, 13, 12, 9, 12, 9, 9, 14, 12, 7, 15, 12, 6, 15, 13, 6, 21, 12, 12, 13, 6, 11, 15, 15, 9, 14, 12, 8, 24, 10, 9

Offset: 0

Author

Joerg Arndt, Mar 21 2014

Keywords

• nonn

Examples

			The a(33) = 15 such compositions of 33 are:
01:  [ 1 2 3 4 5 6 5 4 3 ]
02:  [ 2 3 4 5 6 7 6 ]
03:  [ 3 4 5 6 5 4 3 2 1 ]
04:  [ 3 4 5 6 7 8 ]
05:  [ 4 5 6 7 6 5 ]
06:  [ 5 6 7 6 5 4 ]
07:  [ 5 6 7 8 7 ]
08:  [ 6 7 6 5 4 3 2 ]
09:  [ 7 8 7 6 5 ]
10:  [ 8 7 6 5 4 3 ]
11:  [ 10 11 12 ]
12:  [ 12 11 10 ]
13:  [ 16 17 ]
14:  [ 17 16 ]
15:  [ 33 ]
G.f. = 1 + x + x^2 + 3*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 3*x^8 + 6*x^9 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Dandan Chen and Rong Chen, Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions, arXiv:2107.04809 [math.NT], 2021.

Crossrefs

Cf. A001522, A001523, A005169, A034297, A059618, A238870, A238871, A130695.

Programs

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], If[ OddQ[n], 1, 1/3] Length @ FindInstance[ {x >= 0, y >= 0, z >= 0, x y + y z + z x + x + y + z + 1 == n}, {x, y, z}, Integers, 10^9]]; (* Michael Somos, Jul 04 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ FindInstance[ {1 <= y <= n, 1 <= x <= y, 1 <= z <= y, y^2 + (x - x^2 + z - z^2) / 2 == n}, {x, y, z}, Integers, 10^9]]; (* Michael Somos, Jul 04 2015 *)

PARI

\\ generate the compositions
a(n)=
{
    if ( n==0, return(1) );
    my( ret=0 );
    my( as, ts );
    for (f=1, n,  \\ first part
        as = 0;
        for (p=f, n, \\ numper of parts in rising half
            as += p; \\ ascending sum
            if ( as > n, break() );
            if ( as == n,  ret+=1;  break() );
            ts = as;  \\ total sum
            forstep (q=p-1, 1, -1,
                ts += q;  \\ descending sum
                if ( ts > n, break() );
                if ( ts == n,  ret+=1;  break() );
            );
        );
    );
    return( ret );
}
v=vector(100,n,a(n-1))

Formula

a(2*n) = A130695(2*n) / 3 if n>0. a(2*n + 1) = A130695(2*n + 1) = 3 * H(8*n + 3), where H is the Hurwitz class number, if n>0. - Michael Somos, Jul 04 2015