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 A040039 First differences of A033485; also A033485 with terms repeated. 13
 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. Also the number of unlabeled binary rooted trees with 2*n + 3 nodes in which the two branches directly under any given non-leaf node are either equal or at least one of them is a leaf. - Gus Wiseman, Oct 08 2018 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 Using offset 1: a(1) = 1; a(n even) = a(n-1); a(n odd) = a(n-1) + a((n-1)/2). - Gus Wiseman, Oct 08 2018 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 ]     [ 11 6 2 ]     [ 11 7 1 ]     [ 11 8 ]     [ 12 4 2 1 ]     [ 12 4 3 ]     [ 12 5 2 ]     [ 12 6 1 ]     [ 12 7 ]     [ 13 4 2 ]     [ 13 5 1 ]     [ 13 6 ]     [ 14 3 2 ]     [ 14 4 1 ]     [ 14 5 ]     [ 15 3 1 ]     [ 15 4 ]     [ 16 2 1 ]     [ 16 3 ]     [ 17 2 ]     [ 18 1 ]     [ 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) From Gus Wiseman, Oct 08 2018: (Start) The a(-1) = 1 through a(4) = 3 semichiral binary rooted trees:   o  (oo)  (o(oo))  ((oo)(oo))  (o((oo)(oo)))  ((o(oo))(o(oo)))                     (o(o(oo)))  (o(o(o(oo))))  (o(o((oo)(oo))))                                                (o(o(o(o(oo))))) (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], . # 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 jsum(r[k], k=j+1..nr) do j:=j+1; od; # gives A040039 #while 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 MATHEMATICA T[n_, m_] := T[n, m] = Sum[Binomial[n-2k-1, n-2k-m] Sum[Binomial[m, i] T[k, i], {i, 1, k}], {k, 0, (n-m)/2}] + Binomial[n-1, n-m]; a[n_] := T[n+1, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 27 2018, after Vladimir Kruchinin *) 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. Cf. A001190, A003238, A111299, A292050, A317712, A320222, A320230, A320271. Sequence in context: A283529 A064986 A029019 * A008667 A239880 A240862 Adjacent sequences:  A040036 A040037 A040038 * A040040 A040041 A040042 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 10 12:42 EST 2019. Contains 329895 sequences. (Running on oeis4.)