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A040039 First differences of A033485; also A033485 with terms repeated. 12
1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 13, 18, 18, 23, 23, 30, 30, 37, 37, 47, 47, 57, 57, 70, 70, 83, 83, 101, 101, 119, 119, 142, 142, 165, 165, 195, 195, 225, 225, 262, 262, 299, 299, 346, 346, 393, 393, 450, 450, 507, 507, 577, 577, 647, 647, 730, 730, 813, 813, 914, 914, 1015, 1015, 1134, 1134, 1253, 1253, 1395, 1395 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Apparently a(n) = number of partitions (p_1, p_2, ..., p_k) of n+1, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i > p_{i+1}+...+p_k. - John McKay (mac(AT)mathstat.concordia.ca), Mar 06 2009

Comment from John McKay confirmed in paper by Bessenrodt, Olsson, and Sellers. Such partitions are called "strongly decreasing" partitions in the paper, see the function s(n) therein.

LINKS

Joerg Arndt, Table of n, a(n) for n = 0..500

Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, Unique path partitions: Characterization and Congruences, arXiv:1107.1156 [math.CO], 2011-2012.

J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. - From N. J. A. Sloane, Feb 03 2013

FORMULA

Let T(x) be the g.f, then T(x) = 1 + x/(1-x)*T(x^2) = 1 + x/(1-x) * ( 1 + x^2/(1-x^2) * ( 1 + x^4/(1-x^4) * ( 1 + x^8/(1-x^8) *(...) ))). [Joerg Arndt, May 11 2010]

From Joerg Arndt, Oct 02 2013: (Start)

G.f.: sum(k>=1, x^(2^k-1) / prod(j=0..k-1, 1-x^(2^k) ) ) [Bessenrodt/Olsson/Sellers].

G.f.: 1/(2*x^2) * ( 1/prod(k>=0, 1 - x^(2^k) ) - (1 + x) ).

a(n) = 1/2 * A018819(n+2).

(End)

a(n) = T(n+1,1), where T(n,m)=sum(k..0,(n-m)/2, binomial(n-2*k-1,n-2*k-m)*sum(i=1..k, binomial(m,i)*T(k,i)))+binomial(n-1,n-m). - Vladimir Kruchinin, Mar 19 2015

EXAMPLE

From Joerg Arndt, Dec 17 2012: (Start)

The a(19-1)=30 strongly decreasing partitions of 19 are (in lexicographic order)

[ 1]    [ 10 5 3 1 ]

[ 2]    [ 10 5 4 ]

[ 3]    [ 10 6 2 1 ]

[ 4]    [ 10 6 3 ]

[ 5]    [ 10 7 2 ]

[ 6]    [ 10 8 1 ]

[ 7]    [ 10 9 ]

[ 8]    [ 11 5 2 1 ]

[ 9]    [ 11 5 3 ]

[10]    [ 11 6 2 ]

[11]    [ 11 7 1 ]

[12]    [ 11 8 ]

[13]    [ 12 4 2 1 ]

[14]    [ 12 4 3 ]

[15]    [ 12 5 2 ]

[16]    [ 12 6 1 ]

[17]    [ 12 7 ]

[18]    [ 13 4 2 ]

[19]    [ 13 5 1 ]

[20]    [ 13 6 ]

[21]    [ 14 3 2 ]

[22]    [ 14 4 1 ]

[23]    [ 14 5 ]

[24]    [ 15 3 1 ]

[25]    [ 15 4 ]

[26]    [ 16 2 1 ]

[27]    [ 16 3 ]

[28]    [ 17 2 ]

[29]    [ 18 1 ]

[30]    [ 19 ]

The a(20-1)=30 strongly decreasing partitions of 20 are obtained by adding 1 to the first part in each partition in the list.

(End)

MAPLE

# For example, the five partitions of 4, written in nonincreasing order, are

# [1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4].

# Only the last two satisfy the condition, and a(3)=2.

# The Maple program below verifies this for small values of n.

with(combinat); N:=8; a:=array(1..N); c:=array(1..N);

for n from 1 to N do p:=partition(n); np:=nops(p); t:=0;

for s to np do r:=p[s]; r:=sort(r, `>`); nr:=nops(r); j:=1;

while j<nr and r[j]>sum(r[k], k=j+1..nr) do j:=j+1; od; # gives A040039

#while j<nr and r[j]>= sum(r[k], k=j+1..nr) do j:=j+1; od; # gives A018819

if j=nr then t:=t+1; fi od; a[n]:=t; od;

# John McKay

PROG

(PARI) /* compute as "A033485 with terms repeated" */

b(n) = if(n<2, 1, b(floor(n/2))+b(n-1));  /* A033485 */

a(n) = b(n\2+1); /* note different offsets */

for(n=0, 99, print1(a(n), ", ")); /* Joerg Arndt, Jan 21 2011 */

(Maxima)

T(n, m):=sum(binomial(n-2*k-1, n-2*k-m)*sum(binomial(m, i)*T(k, i), i, 1, k), k, 0, (n-m)/2)+binomial(n-1, n-m);

makelist(T(n+1, 1), n, 0, 40); /* Vladimir Kruchinin, Mar 19 2015 */

CROSSREFS

Cf. A000123, A018819.

Cf. A018819, A088567, A089054.

Sequence in context: A283529 A064986 A029019 * A008667 A239880 A240862

Adjacent sequences:  A040036 A040037 A040038 * A040040 A040041 A040042

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane and J. H. Conway

STATUS

approved

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Last modified June 26 02:24 EDT 2017. Contains 288749 sequences.