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%I #37 Mar 27 2021 22:46:09
%S 1,1,2,2,3,3,5,6,6,9,11,10,15,18,19,24,31,29,40,44,51,56,72,69,90,97,
%T 114,125,154,151,192,207,237,255,304,314,377,401,457,493,573,596,698,
%U 750,845,905,1034,1104,1255,1354,1507,1624,1817,1955,2178,2357,2605,2794,3077,3380
%N Number of partitions of n into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a3, ..., a_{m-1} - a_m are different.
%C In other words, a(n) is the number of strict integer partitions of n with distinct first differences. - _Gus Wiseman_, Mar 25 2021
%H Fausto A. C. Cariboni, <a href="/A320347/b320347.txt">Table of n, a(n) for n = 1..500</a> (terms 1..100 from Seiichi Manyama)
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e n = 9
%e [9] ooooooooo
%e ------------------------------------
%e [8, 1] *******o a_1 - a_2 = 7.
%e oooooooo
%e ------------------------------------
%e [7, 2] *****oo a_1 - a_2 = 5.
%e ooooooo
%e ------------------------------------
%e [6, 3] ***ooo a_1 - a_2 = 3.
%e oooooo
%e ------------------------------------
%e [6, 2, 1] *o a_2 - a_3 = 1.
%e ****oo a_1 - a_2 = 4.
%e oooooo
%e ------------------------------------
%e [5, 4] *oooo a_1 - a_2 = 1.
%e ooooo
%e ------------------------------------
%e a(9) = 6.
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[#]&]],{n,0,30}] (* _Gus Wiseman_, Mar 27 2021 *)
%Y Cf. A000009, A320348.
%Y The equal instead of distinct version is A049980.
%Y The non-strict version is A325325 (ranking: A325368).
%Y The non-strict ordered version is A325545.
%Y The version for first quotients is A342520 (non-strict: A342514).
%Y Cf. A003114, A003242, A005117, A238710, A342530.
%K nonn
%O 1,3
%A _Seiichi Manyama_, Oct 11 2018
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