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A024786 Number of 2's in all partitions of n. 39
0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, 295, 389, 526, 686, 911, 1176, 1538, 1968, 2540, 3223, 4115, 5181, 6551, 8191, 10269, 12756, 15873, 19598, 24222, 29741, 36532, 44624, 54509, 66261, 80524, 97446, 117862, 142029, 171036, 205290, 246211 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also number of partitions of n-1 with a distinguished part different from all the others. [Comment corrected by Emeric Deutsch, Aug 13 2008]

In general the number of times that j appears in the partitions of n equals Sum_{k<n, k = n (mod j)} P(k). In particular this gives a formula for a(n), A024787, ..., A024794, for j = 2,...,10; it generalizes the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002

Equals row sums of triangle A173238. [Gary W. Adamson, Feb 13 2010]

The sums of two successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of second largest and the sum of third largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

Number of singletons in all partitions of n-1. A singleton in a partition is a part that occurs exactly once. Example: a(5) = 4 because in the partitions of 4, namely [1,1,1,1], [1,1,2'], [2,2], [1',3'], [4'] we have 4 singletons (marked by '). - Emeric Deutsch, Sep 12 2016.

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.

E. Deutsch et al., Problem 11237, Amer. Math. Monthly, 115 (No. 7, 2008), 666-667. [From Emeric Deutsch, Aug 13 2008]

FORMULA

a(n) = Sum_{k=1..floor(n/2)} A000041(n-2k). - Christian G. Bower, Jun 22 2000

a(n) = Sum_{k<n, k = n (mod 2)} P(k), P(k) = number of partitions of k as in A000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002

G.f.: x^2/((1-x)*(1-x^2)^2))*Product_{j>=3} 1/(1-x^j) from Riordan reference second term, last eq.

a(n) = A006128(n-1) - A194452(n-1). - Omar E. Pol, Nov 20 2011

a(n) = A181187(n,2) - A181187(n,3). - Omar E. Pol, Oct 25 2012

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2) * Pi * sqrt(n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 433*Pi^2/6912)/n). - Vaclav Kotesovec, Mar 07 2016, extended Nov 05 2016

a(n) = Sum_{k} k * A116595(n-1,k). - Emeric Deutsch, Sep 12 2016

EXAMPLE

Contribution from Omar E. Pol, Oct 25 2012 (Start):

For n = 7 we have:

--------------------------------------

.                             Number

Partitions of 7               of 2's

--------------------------------------

7 .............................. 0

4 + 3 .......................... 0

5 + 2 .......................... 1

3 + 2 + 2 ...................... 2

6 + 1 .......................... 0

3 + 3 + 1 ...................... 0

4 + 2 + 1 ...................... 1

2 + 2 + 2 + 1 .................. 3

5 + 1 + 1 ...................... 0

3 + 2 + 1 + 1 .................. 1

4 + 1 + 1 + 1 .................. 0

2 + 2 + 1 + 1 + 1 .............. 2

3 + 1 + 1 + 1 + 1 .............. 0

2 + 1 + 1 + 1 + 1 + 1 .......... 1

1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0

------------------------------------

.  24 - 13 =                    11

.

The difference between the sum of the second column and the sum of the third column of the set of partitions of 7 is 24 - 13 = 11 and equals the number of 2's in all partitions of 7, so a(7) = 11.

(End)

MAPLE

b:= proc(n, i) option remember; local f, g;

      if n=0 or i=1 then [1, 0]

    else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));

         [f[1]+g[1], f[2]+g[2]+`if`(i=2, g[1], 0)]

      fi

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=1..50);  # Alois P. Heinz, May 18 2012

MATHEMATICA

Table[ Count[ Flatten[ IntegerPartitions[n]], 2], {n, 1, 50} ]

b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, 0}, f = b[n, i - 1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i == 2, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-Fran├žois Alcover, Sep 22 2015, after Alois P. Heinz *)

Join[{0}, (1/((1 - x^2) QPochhammer[x]) + O[x]^50)[[3]]] (* Vladimir Reshetnikov, Nov 22 2016 *)

Table[Sum[(1 + (-1)^k)/2 * PartitionsP[n-k], {k, 2, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 27 2017 *)

CROSSREFS

Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A173238.

Column 2 of A060244.

First differences of A000097.

Sequence in context: A212548 A212549 A212550 * A299069 A097497 A279328

Adjacent sequences:  A024783 A024784 A024785 * A024787 A024788 A024789

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)