cp's OEIS Frontend

A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.

%I A100818 #38 Apr 25 2025 21:12:42
%S A100818 1,2,1,4,3,8,7,15,15,27,29,48,53,82,94,137,160,225,265,362,430,572,
%T A100818 683,892,1066,1370,1640,2078,2487,3117,3725,4624,5519,6791,8092,9885,
%U A100818 11752,14263,16922,20416,24167,29007,34254,40921,48213,57345,67409,79864
%N A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.
%C A100818 Note that this is very similar to the "crank" of Andrews and Garvan. The number of partitions pi with P(pi) odd is the given sequence.
%C A100818 The sequence is the same as A087787 except for the value of a(1) (this was established by George Andrews, Jan 18 2005). If "even" is replace by "odd" in the definition of the sequence, the new sequence is almost identical except for two values and a shift to the right.
%C A100818 Also, positive integers of A182712. a(n) is also the number of 2's in the n-th row that contain a 2 as a part in the triangle of A138121 (note that rows 1 and 3 do not contain a 2 as a part). - _Omar E. Pol_, Nov 28 2010
%H A100818 G. C. Greubel, <a href="/A100818/b100818.txt">Table of n, a(n) for n = 1..1000</a>
%H A100818 G. E. Andrews and F. Garvan, <a href="http://dx.doi.org/10.1090/S0273-0979-1988-15637-6">Dyson's crank of a partition</a>, Bull. Amer. Math. Soc., 18 (1988), 167-171.
%F A100818 G.f.: x+(1/(1+x))* Product_{n>=1}(1/(1-x^n)). [corrected by _Vaclav Kotesovec_, Aug 29 2019]
%F A100818 a(n) = A000041(n) - a(n-1), for n>2. - _Jon Maiga_, Aug 29 2019 [corrected by _Vaclav Kotesovec_, Aug 29 2019]
%F A100818 a(n) = a(n-2) + A000041(n-1) - A000041(n-2), for n>=3. - _Vaclav Kotesovec_, Aug 29 2019
%e A100818 a(3)=1 because P(3)=3, P(2 1)=1 and P(1 1 1)=0.
%t A100818 Rest[ CoefficientList[ Series[x + 1/(1 + x) Product[1/(1 - x^n), {n, 50}], {x, 0, 50}], x]] (* _Robert G. Wilson v_, Feb 11 2005 *)
%Y A100818 Cf. A000041, A087787, A135010, A138121, A182712.
%K A100818 nonn
%O A100818 1,2
%A A100818 _David S. Newman_, Jan 13 2005
%E A100818 More terms from _Robert G. Wilson v_, Feb 11 2005