A333290 Irregular triangle read by rows: coefficients b_{r,j} (r>=1, j>=0) arising from an expansion of the partition function.

1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 0, 2, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 1, 1, -1, -1, -1, 0, 0, 1, 1, -1, 1, -1, -1, 0, 0, 1, 0, 2, 0, -1, -1, -1, -1, 0, 2, 0, 1, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 0, 1, 1, 0, -1, -1, -2, 0, 0, 0, 2, 1, 1, 0, -1, -1, 0, -1, 0, 0, 1, 1, -1, 1, -1, -1, 0, 0, 1, 0, 1, 0, 1, 0, -1, -2, -1, 0, -1, 1

Offset: 1

Author

N. J. A. Sloane, Mar 20 2020

Keywords

• sign
• tabf

Comments

Is this (apart from offset) the same as A231599? - R. J. Mathar, Mar 21 2020

Examples

			Triangle begins:
1,
1,-1,
1,-1,-1,1,
1,-1,-1,0,1,1,-1,
1,-1,-1,0,0,2,0,0,-1,-1,1,
1,-1,-1,0,0,1,1,1,-1,-1,-1,0,0,1,1,-1,
...
		

Links

• Mircea Merca and Maxie D. Schmidt, The partition function p(n) in terms of the classical Möbius function, Ramanujan J (2019) 49:87-96.

Crossrefs

Cf. A333289.

Programs

Maple

A333290 := proc(r,j)
    if r < 1 then
        0 ;
    elif r = 1 then
        if j= 0 then
            1;
        else
            0 ;
        end if;
    elif j < r-1 then
        procname(r-1,j) ;
    else
        procname(r-1,j) -procname(r-1,j-r+1) ;;
    end if;
end proc: # R. J. Mathar, Mar 21 2020

Mathematica

b[r_, j_] := b[r, j] = Which[r < 1, 0, r == 1, If[j == 0, 1, 0], j < r-1, b[r-1, j], True, b[r-1, j] - b[r-1, j-r+1]];
Table[b[r, j], {r, 1, 9}, {j, 0, r(r-1)/2}] // Flatten (* Jean-François Alcover, Apr 29 2023, after R. J. Mathar *)