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A182714 Number of 4's in the last section of the set of partitions of n. 22
0, 0, 0, 1, 0, 1, 1, 3, 2, 5, 5, 10, 10, 17, 19, 31, 34, 51, 60, 86, 100, 139, 165, 223, 265, 349, 418, 543, 648, 827, 992, 1251, 1495, 1866, 2230, 2758, 3289, 4033, 4803, 5852, 6949, 8411, 9973, 12005, 14194, 17002, 20060, 23919, 28153, 33426, 39256, 46438 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Zero together with the first differences of A024788.

Also number of 4's in all partitions of n that do not contain 1 as a part.

a(n) is the number of partitions of n such that m(1) < m(3), where m = multiplicity; e.g., a(7) counts these 3 partitions: [4, 3], [3, 3, 1], [3, 2, 2]. - Clark Kimberling, Apr 01 2014

The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

LINKS

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

FORMULA

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3) + a(n+4), n >= 0. - Omar E. Pol, Feb 04 2012

EXAMPLE

a(8) = 3 counts the 4's in 8 = 4+4 = 4+2+2. The 4's in 8 = 4+3+1 = 4+2+1+1 = 4+1+1+1+1 do not count.

From Omar E. Pol, _Oct 25 2012_ (Start):

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

Last section                   Number

of the set of                    of

partitions of 8                 4's

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

8 .............................. 0

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

5 + 3 .......................... 0

6 + 2 .......................... 0

3 + 3 + 2 ...................... 0

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

2 + 2 + 2 + 2 .................. 0

.   1 .......................... 0

.       1 ...................... 0

.       1 ...................... 0

.           1 .................. 0

.       1 ...................... 0

.           1 .................. 0

.           1 .................. 0

.               1 .............. 0

.           1 .................. 0

.               1 .............. 0

.               1 .............. 0

.                   1 .......... 0

.                   1 .......... 0

.                       1 ...... 0

.                           1 .. 0

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

.           6 - 3 =              3

.

In the last section of the set of partitions of 8 the difference between the sum of the fourth column and the sum of the fifth column is 6 - 3 = 3 equaling the number of 4's, so a(8) = 3 (see also A024788).

(End)

MAPLE

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

      if n=0 then [1, 0]

    elif i<2 then [0, 0]

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

         [g[1]+h[1], g[2]+h[2]+`if`(i=4, h[1], 0)]

      fi

    end:

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

seq (a(n), n=1..70);  # Alois P. Heinz, Mar 19 2012

MATHEMATICA

MM  (See A240058).  - Clark Kimberling, Apr 01 2014

PROG

(Sage) A182714 = lambda n: sum(list(p).count(4) for p in Partitions(n) if 1 not in p)

CROSSREFS

Column 4 of A194812.

Cf. A015739, A024788, A135010, A138121, A182703, A182712, A182713, A240058.

Sequence in context: A186545 A008623 A035546 * A198755 A134237 A227192

Adjacent sequences:  A182711 A182712 A182713 * A182715 A182716 A182717

KEYWORD

nonn

AUTHOR

Omar E. Pol, Nov 13 2011

STATUS

approved

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Last modified April 23 13:12 EDT 2014. Contains 240927 sequences.