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 A179080 Number of partitions of n into distinct parts where all differences between consecutive parts are odd. 3
 1, 1, 1, 2, 1, 3, 2, 4, 2, 6, 4, 7, 5, 9, 8, 12, 10, 14, 15, 17, 19, 22, 26, 26, 32, 32, 42, 40, 52, 48, 66, 59, 79, 73, 98, 89, 118, 108, 143, 133, 170, 160, 204, 194, 241, 236, 286, 283, 336, 339, 396, 407, 464, 483, 544, 575, 634, 681, 740, 803, 862, 944, 1001, 1110, 1162, 1296, 1348, 1512, 1561, 1760, 1805 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 FORMULA G.f.: sum(n>=0, x^(n*(n+1)/2) / prod(k=1..n+1, 1-x^(2*k) ) ). - Joerg Arndt, Jan 29 2011 a(n) = A179049(n) + A218355(n). - Joerg Arndt, Oct 27 2012 EXAMPLE From Joerg Arndt, Oct 27 2012:  (Start) The a(18) = 15 such partitions of 18 are: [ 1]  1 2 3 12 [ 2]  1 2 5 10 [ 3]  1 2 7 8 [ 4]  1 2 15 [ 5]  1 4 5 8 [ 6]  1 4 13 [ 7]  1 6 11 [ 8]  1 8 9 [ 9]  2 3 4 9 [10]  2 3 6 7 [11]  3 4 5 6 [12]  3 4 11 [13]  3 6 9 [14]  5 6 7 [15]  18 (End) MAPLE b:= proc(n, i) option remember; `if`(n=0, 1,       `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))     end: a:= n-> `if`(n=0, 1, b(n, 1)+b(n, 2)): seq(a(n), n=0..100);  # Alois P. Heinz, Nov 08 2012; revised Feb 24 2020 MATHEMATICA b[n_, i_, t_] := b[n, i, t] = If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i <= n && Mod[i, 2] != t, b[n-i, i-1, Mod[i, 2]], 0]]]; a[n_] := If[n==0, 1, Sum[b[n-i, i-1, Mod[i, 2]], {i, 1, n}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *) PROG (Sage) def A179080(n):     odd_diffs = lambda x: all(abs(d) % 2 == 1 for d in differences(x))     satisfies = lambda p: not p or odd_diffs(p)     def count(pred, iter): return sum(1 for item in iter if pred(item))     return count(satisfies, Partitions(n, max_slope=-1)) print([A179080(n) for n in range(0, 20)]) # show first terms (Sage) # Alternative after Alois P. Heinz: def A179080(n):     @cached_function     def h(n, k):         if n == 0: return 1         if k  > n: return 0         return h(n, k+2) + h(n-k, k+1)     return h(n, 1) + h(n, 2) if n > 0 else 1 print([A179080(n) for n in range(71)]) # Peter Luschny, Feb 25 2020 (PARI) N=66; x='x+O('x^N); gf = sum(n=0, N, x^(n*(n+1)/2) / prod(k=1, n+1, 1-x^(2*k) ) ); Vec( gf ) /* Joerg Arndt, Jan 29 2011 */ CROSSREFS Cf. A179049 (odd differences and odd minimal part). Cf. A189357 (even differences, distinct parts), A096441 (even differences). Cf. A000009 (partitions of 2*n with even differences and even minimal part). Sequence in context: A308308 A024162 A334677 * A294199 A078658 A307719 Adjacent sequences:  A179077 A179078 A179079 * A179081 A179082 A179083 KEYWORD nonn AUTHOR Joerg Arndt, Jan 04 2011 STATUS approved

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)