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%I #73 Jul 27 2021 21:19:08
%S 1,2,3,6,8,15,19,32,42,64,83,124,157,224,288,395,502,679,854,1132,
%T 1422,1847,2307,2968,3677,4671,5772,7251,8908,11110,13572,16792,20439,
%U 25096,30414,37138,44798,54389,65386,78959,94558,113687,135646,162375,193133
%N First differences of A006128.
%C Number of parts in the last section of the set of partitions of n (see A135010, A138121).
%C Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - _Omar E. Pol_, Nov 07 2011
%C From _Omar E. Pol_, Feb 16 2021: (Start)
%C Convolution of A341062 and A000041.
%C Convolution of A000005 and A002865.
%C a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
%C a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)
%H Alois P. Heinz, <a href="/A138137/b138137.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A006128(n) - A006128(n-1).
%F a(n) = A000041(n-1) + A138135(n). - _Omar E. Pol_, Nov 07 2011
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 21 2016
%F G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - _Ilya Gutkovskiy_, Apr 04 2017
%e From _Omar E. Pol_, Feb 19 2012: (Start)
%e Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
%e . 6
%e . 3+3
%e . 4+2
%e . 2+2+2
%e . 5 1
%e . 3+2 1
%e . 4 1 1
%e . 2+2 1 1
%e . 3 1 1 1
%e . 2 1 1 1 1
%e . 1 1 1 1 1 1
%e . --- ----- ------- --------- ----------- --------------
%e . 1, 2, 3, 6, 8, 15,
%e ...
%e Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)
%e From _Omar E. Pol_, Aug 16 2013: (Start)
%e The geometric model looks like this:
%e . _ _ _ _ _ _
%e . |_ _ _ _ _ _|
%e . |_ _ _|_ _ _|
%e . |_ _ _ _|_ _|
%e . _ _ _ _ _ |_ _|_ _|_ _|
%e . |_ _ _ _ _| |_|
%e . _ _ _ _ |_ _ _|_ _| |_|
%e . |_ _ _ _| |_| |_|
%e . _ _ _ |_ _|_ _| |_| |_|
%e . _ _ |_ _ _| |_| |_| |_|
%e . _ |_ _| |_| |_| |_| |_|
%e . |_| |_| |_| |_| |_| |_|
%e .
%e . 1 2 3 6 8 15
%e .
%e (End)
%e On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - _Omar E. Pol_, Jul 27 2021
%p b:= proc(n, i) option remember; local f, g;
%p if n=0 then [1, 0]
%p elif i<1 then [0, 0]
%p elif i>n then b(n, i-1)
%p else f:= b(n, i-1); g:= b(n-i, i);
%p [f[1]+g[1], f[2]+g[2] +g[1]]
%p fi
%p end:
%p a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
%p seq(a(n), n=1..50); # _Alois P. Heinz_, Feb 19 2012
%t b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Mar 03 2014, after _Alois P. Heinz_ *)
%t Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* _Robert Price_, May 01 2020 *)
%Y Column 1 of A207031.
%Y Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.
%K easy,nonn
%O 1,2
%A _Omar E. Pol_, Mar 18 2008
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