login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A276429 Number of partitions of n containing no part i of multiplicity i. 17
1, 0, 2, 2, 3, 5, 8, 9, 16, 19, 29, 36, 53, 65, 92, 115, 154, 195, 257, 318, 419, 516, 663, 821, 1039, 1277, 1606, 1963, 2441, 2978, 3675, 4454, 5469, 6603, 8043, 9688, 11732, 14066, 16963, 20260, 24310, 28953, 34586, 41047, 48857, 57802, 68528, 80862, 95534, 112388, 132391 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The Heinz numbers of these partitions are given by A325130. - Gus Wiseman, Apr 02 2019

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..12782 (terms 0..5000 from Alois P. Heinz)

FORMULA

a(n) = A276427(n,0).

G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^{i^2}).

EXAMPLE

a(4) = 3 because we have [1,1,1,1], [1,1,2], and [4]; the partitions [1,3], [2,2] do not qualify.

From Gus Wiseman, Apr 02 2019: (Start)

The a(2) = 2 through a(7) = 9 partitions:

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

  (11)  (111)  (211)   (32)     (33)      (43)

               (1111)  (311)    (42)      (52)

                       (2111)   (222)     (511)

                       (11111)  (411)     (3211)

                                (3111)    (4111)

                                (21111)   (31111)

                                (111111)  (211111)

                                          (1111111)

(End)

MAPLE

g := product(1/(1-x^i)-x^(i^2), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(`if`(i=j, 0, b(n-i*j, i-1)), j=0..n/i)))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..60);  # Alois P. Heinz, Sep 19 2016

MATHEMATICA

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n][[1]], {n, 0, 60}] (* Jean-Fran├žois Alcover, Nov 28 2016 after Alois P. Heinz's Maple code for A276427 *)

Table[Length[Select[IntegerPartitions[n], And@@Table[Count[#, i]!=i, {i, Union[#]}]&]], {n, 0, 30}] (* Gus Wiseman, Apr 02 2019 *)

CROSSREFS

Cf. A052335, A087153, A114639, A115584, A117144, A276427, A324572, A325130, A325131.

Sequence in context: A114990 A241421 A157176 * A111181 A267419 A076777

Adjacent sequences:  A276426 A276427 A276428 * A276430 A276431 A276432

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 19 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 16 06:18 EST 2019. Contains 330016 sequences. (Running on oeis4.)