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A040039
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First differences of A033485; also A033485 with terms repeated.
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6
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| Comment from John MCKAY (mckay(AT)encs.concordia.ca), Mar 06 2009 (Start): 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.
Comment from John MCKAY confirmed in paper by Bessenrodt, Olsson, and Sellers. (Such partitions are called "strongly decreasing" partitions in the paper by Bessenrodt, Olsson, and Sellers; see the function s(n) therein.)
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LINKS
| Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, Unique path partitions: Characterization and Congruences, arXiv preprint
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FORMULA
| (Conjectured) let T(x) be the g.f, then T(x)=1+x/(1-x)*T(x^2). [Joerg Arndt, May 11 2010]
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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. (End)
(Maple code from John McKay) 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;
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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), ", ")); /* show terms */ /* Joerg Arndt, Jan 21 2011 */
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CROSSREFS
| Cf. A000123, A018819.
Cf. A018819, A088567, A089054.
Sequence in context: A085885 A064986 A029019 * A008667 A177716 A109763
Adjacent sequences: A040036 A040037 A040038 * A040040 A040041 A040042
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)
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