cp's OEIS Frontend

A320347 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.

%I A320347 #37 Mar 27 2021 22:46:09
%S A320347 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 A320347 114,125,154,151,192,207,237,255,304,314,377,401,457,493,573,596,698,
%U A320347 750,845,905,1034,1104,1255,1354,1507,1624,1817,1955,2178,2357,2605,2794,3077,3380
%N A320347 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 A320347 In other words, a(n) is the number of strict integer partitions of n with distinct first differences. - _Gus Wiseman_, Mar 25 2021
%H A320347 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 A320347 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A320347 n = 9
%e A320347 [9]        ooooooooo
%e A320347 ------------------------------------
%e A320347 [8, 1]      *******o  a_1 - a_2 = 7.
%e A320347             oooooooo
%e A320347 ------------------------------------
%e A320347 [7, 2]       *****oo  a_1 - a_2 = 5.
%e A320347              ooooooo
%e A320347 ------------------------------------
%e A320347 [6, 3]        ***ooo  a_1 - a_2 = 3.
%e A320347               oooooo
%e A320347 ------------------------------------
%e A320347 [6, 2, 1]         *o  a_2 - a_3 = 1.
%e A320347               ****oo  a_1 - a_2 = 4.
%e A320347               oooooo
%e A320347 ------------------------------------
%e A320347 [5, 4]         *oooo  a_1 - a_2 = 1.
%e A320347                ooooo
%e A320347 ------------------------------------
%e A320347 a(9) = 6.
%t A320347 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[#]&]],{n,0,30}] (* _Gus Wiseman_, Mar 27 2021 *)
%Y A320347 Cf. A000009, A320348.
%Y A320347 The equal instead of distinct version is A049980.
%Y A320347 The non-strict version is A325325 (ranking: A325368).
%Y A320347 The non-strict ordered version is A325545.
%Y A320347 The version for first quotients is A342520 (non-strict: A342514).
%Y A320347 Cf. A003114, A003242, A005117, A238710, A342530.
%K A320347 nonn
%O A320347 1,3
%A A320347 _Seiichi Manyama_, Oct 11 2018