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A333290 Irregular triangle read by rows: coefficients b_{r,j} (r>=1, j>=0) arising from an expansion of the partition function. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,20
COMMENTS
Is this (apart from offset) the same as A231599? - R. J. Mathar, Mar 21 2020
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.
EXAMPLE
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,
...
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 *)
CROSSREFS
Cf. A333289.
Sequence in context: A241035 A140698 A231599 * A321924 A124764 A151899
KEYWORD
sign,tabf
AUTHOR
N. J. A. Sloane, Mar 20 2020
STATUS
approved

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)