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

1,2

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

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.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A268630 A087787 A182712 * A005291 A106624 A028297

Adjacent sequences: A100815 A100816 A100817 * A100819 A100820 A100821

Keyword

nonn

Author

David S. Newman, Jan 13 2005

Extensions

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

Status

approved