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.

1, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409, 79864

Offset: 1

Author

David S. Newman, Jan 13 2005

Keywords

• nonn

Comments

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.

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.

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

Examples

			a(3)=1 because P(3)=3, P(2 1)=1 and P(1 1 1)=0.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.

Crossrefs

Cf. A000041, A087787, A135010, A138121, A182712.

Programs

Mathematica

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 *)

Formula

G.f.: x+(1/(1+x))* Product_{n>=1}(1/(1-x^n)). [corrected by Vaclav Kotesovec, Aug 29 2019]

a(n) = A000041(n) - a(n-1), for n>2. - Jon Maiga, Aug 29 2019 [corrected by Vaclav Kotesovec, Aug 29 2019]

a(n) = a(n-2) + A000041(n-1) - A000041(n-2), for n>=3. - Vaclav Kotesovec, Aug 29 2019

Extensions

More terms from Robert G. Wilson v, Feb 11 2005