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A064174 Number of partitions of n with nonnegative rank. 19
1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, 42, 56, 73, 96, 125, 161, 207, 265, 336, 426, 536, 672, 840, 1046, 1296, 1603, 1975, 2425, 2970, 3628, 4417, 5367, 6503, 7861, 9482, 11412, 13702, 16423, 19642, 23447, 27938, 33231, 39453, 46767, 55342, 65386, 77135 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The rank of a partition is the largest summand minus the number of summands.

This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - Wouter Meeussen, Mar 29 2013

Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014

From Gus Wiseman, Mar 09 2019: (Start)

Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:

  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)

            (21)  (22)  (32)   (33)   (43)    (44)

                  (31)  (41)   (42)   (52)    (53)

                        (311)  (51)   (61)    (62)

                               (321)  (322)   (71)

                               (411)  (331)   (332)

                                      (421)   (422)

                                      (511)   (431)

                                      (4111)  (521)

                                              (611)

                                              (4211)

                                              (5111)

Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:

  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)

             (111)  (211)   (311)    (321)     (331)      (2222)

                    (1111)  (2111)   (2211)    (2221)     (3221)

                            (11111)  (3111)    (3211)     (3311)

                                     (21111)   (4111)     (4211)

                                     (111111)  (22111)    (22211)

                                               (31111)    (32111)

                                               (211111)   (41111)

                                               (1111111)  (221111)

                                                          (311111)

                                                          (2111111)

                                                          (11111111)

(End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cristina Ballantine, Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.

Rekha Biswal bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n (Mathoverflow)

FORMULA

a(n) = (A000041(n) + A047993(n))/2.

a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004

G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015

EXAMPLE

a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.

MAPLE

f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2), k=1..floor((1+sqrt(24*n+1))/6)):

map(f, [$1..100]); # Robert Israel, Aug 03 2015

MATHEMATICA

Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]

(* also *)

Table[Count[IntegerPartitions[2n], q_/; Last[q]===1 && Max[q-PadRight[Rest[q], Length[q]]]<=1 && Count[First/@Partition[q, 2], _?OddQ]==Count[Last/@Partition[q, 2], _?OddQ]], {n, 16}]

(* also *)

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)

PROG

(PARI) {a(n) = my(A=1); A = sum(m=0, n, x^m*prod(k=1, m, (1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A, n)}

for(n=1, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 03 2015

CROSSREFS

Cf. A063995, A064173.

Row sums of triangle A105806.

Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.

Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.

Sequence in context: A035992 A036003 A027338 * A062121 A094995 A018591

Adjacent sequences:  A064171 A064172 A064173 * A064175 A064176 A064177

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Sep 20 2001

EXTENSIONS

Mathematica programs modified by Clark Kimberling, Feb 12 2014

STATUS

approved

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Last modified February 20 04:24 EST 2020. Contains 332063 sequences. (Running on oeis4.)