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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.

%I A333290 #22 Apr 30 2023 07:11:31
%S A333290 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,
%T A333290 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,
%U A333290 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
%N A333290 Irregular triangle read by rows: coefficients b_{r,j} (r>=1, j>=0) arising from an expansion of the partition function.
%C A333290 Is this (apart from offset) the same as A231599? - _R. J. Mathar_, Mar 21 2020
%H A333290 Mircea Merca and Maxie D. Schmidt, <a href="https://doi.org/10.1007/s11139-017-9988-0">The partition function p(n) in terms of the classical Möbius function</a>, Ramanujan J (2019) 49:87-96.
%e A333290 Triangle begins:
%e A333290 1,
%e A333290 1,-1,
%e A333290 1,-1,-1,1,
%e A333290 1,-1,-1,0,1,1,-1,
%e A333290 1,-1,-1,0,0,2,0,0,-1,-1,1,
%e A333290 1,-1,-1,0,0,1,1,1,-1,-1,-1,0,0,1,1,-1,
%e A333290 ...
%p A333290 A333290 := proc(r,j)
%p A333290     if r < 1 then
%p A333290         0 ;
%p A333290     elif r = 1 then
%p A333290         if j= 0 then
%p A333290             1;
%p A333290         else
%p A333290             0 ;
%p A333290         end if;
%p A333290     elif j < r-1 then
%p A333290         procname(r-1,j) ;
%p A333290     else
%p A333290         procname(r-1,j) -procname(r-1,j-r+1) ;;
%p A333290     end if;
%p A333290 end proc: # _R. J. Mathar_, Mar 21 2020
%t A333290 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]];
%t A333290 Table[b[r, j], {r, 1, 9}, {j, 0, r(r-1)/2}] // Flatten (* _Jean-François Alcover_, Apr 29 2023, after _R. J. Mathar_ *)
%Y A333290 Cf. A333289.
%K A333290 sign,tabf
%O A333290 1,20
%A A333290 _N. J. A. Sloane_, Mar 20 2020