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A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

%I A187219 #63 Feb 17 2023 09:28:42
%S A187219 1,1,1,2,2,4,4,7,8,12,14,21,24,34,41,55,66,88,105,137,165,210,253,320,
%T A187219 383,478,574,708,847,1039,1238,1507,1794,2167,2573,3094,3660,4378,
%U A187219 5170,6153,7245,8591,10087,11914,13959,16424,19196,22519,26252,30701,35717
%N A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.
%C A187219 Essentially the same as A002865, but here a(1) = 1 not 0.
%C A187219 Also number of regions in the last section of the set of partitions of n.
%C A187219 Also number of partitions of n+k that are formed by k+1 sections, k >= 0 (Cf. A194799). - _Omar E. Pol_, Jan 30 2012
%C A187219 For the definition of region see A206437. - _Omar E. Pol_, Aug 13 2013
%C A187219 Partial sums give A000041, n >= 1. - _Omar E. Pol_, Sep 04 2013
%C A187219 Also the number of partitions of n with no parts greater than the number of ones. - _Spencer Miller_, Jan 28 2023
%H A187219 Paolo Xausa, <a href="/A187219/b187219.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H A187219 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%F A187219 a(n) = A083751(n) + 1. - _Omar E. Pol_, Mar 04 2012
%F A187219 a(n) = A002865(n), if n >= 2. - _Omar E. Pol_, Aug 13 2013
%e A187219 From _Omar E. Pol_, Aug 13 2013: (Start)
%e A187219 Illustration of initial terms as number of regions:
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%e A187219 .                    |_ _    |         | |           | |
%e A187219 .             _ _ _  |_ _|_  |         | |           | |
%e A187219 .       _ _  |_ _  |       | |         | |           | |
%e A187219 .   _  |_  |     | |       | |         | |           | |
%e A187219 .  |_|   |_|     |_|       |_|         |_|           |_|
%e A187219 .
%e A187219 .   1     1       1         2           2             4
%e A187219 .
%e A187219 (End)
%t A187219 Join[{1}, Drop[CoefficientList[Series[1 / Product[(1 - x^k)^1, {k, 2, 50}], {x, 0, 50}], x], 2]] (* _Vincenzo Librandi_, Feb 15 2018 *)
%t A187219 A187219[nmax_]:=Join[{1},Differences[PartitionsP[Range[nmax]]]];
%t A187219 A187219[100] (* _Paolo Xausa_, Feb 17 2023 *)
%Y A187219 Cf. A000041, A002865, A083751, A135010, A186114, A193870, A206437, A225600, A225610.
%K A187219 nonn
%O A187219 1,4
%A A187219 _Omar E. Pol_, Dec 09 2011
%E A187219 Better definition from _Omar E. Pol_, Sep 04 2013