This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211978 #50 Mar 20 2020 15:08:44
%S A211978 0,2,6,12,24,40,70,108,172,256,384,550,798,1112,1560,2136,2926,3930,
%T A211978 5288,6996,9260,12104,15798,20412,26348,33702,43044,54588,69090,86906,
%U A211978 109126,136270,169854,210732,260924,321752,396028,485624,594402,725174,883092,1072208
%N A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.
%C A211978 Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.
%C A211978 a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - _Emeric Deutsch_, Oct 07 2016
%C A211978 a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - _Omar E. Pol_, Nov 23 2016
%H A211978 Michael De Vlieger, <a href="/A211978/b211978.txt">Table of n, a(n) for n = 0..3000</a>
%H A211978 Toufik Mansour, Armend Sh. Shabani, <a href="http://www.dmlett.com/archive/DML19_v2_p.65_94.pdf">Enumerations on bargraphs</a>, Discrete Math. Lett. (2019) Vol. 2, 65-94.
%H A211978 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a diagram for A006128</a>
%H A211978 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A211978 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A211978 a(n) = 2*A006128(n).
%F A211978 a(n) = A225600(2*A000041(n)) = A225600(A139582(n)), n >= 1.
%F A211978 a(n) = (Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m)) = 2*Sum_{m=1..p(n)} A194446(m) = 2*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.
%F A211978 The trivariate g.f. G(t,s,x) of the partitions of a nonnegative integer relative to weight (marked by x), number of parts (marked by t), and largest part (marked by s) is G(t,s,x) = Sum_{i>=1} t*s^i*x^i/product_{j=1..i} (1-tx^j). Setting s = t, we obtain the bivariate g.f. of the partitions relative to weight (marked by x) and semiperimeter of the Ferrers board (marked by t). The g.f. of a(n) is g(x) = Sum_{i>=1} ((x^i*(1 + i + Q(x))/R(x)), where Q(x) = sum_{j=1..i} (x^j/(1 - x^j)) and R(x) = product_{j=1..i}(1-x^j). g(x) has been obtained by setting t = 1 in dG(t,t,x))/dt. - _Emeric Deutsch_, Oct 07 2016
%e A211978 Illustration of initial terms as a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
%e A211978 . _ _ _ _ _ _
%e A211978 . _ _ _ |
%e A211978 . _ _ _|_ |
%e A211978 . _ _ | |
%e A211978 . _ _ _ _ _ _ _|_ _|_ |
%e A211978 . _ _ _ | _ _ _ | |
%e A211978 . _ _ _ _ _ _ _|_ | _ _ _|_ | |
%e A211978 . _ _ | _ _ | | _ _ | | |
%e A211978 . _ _ _ _ _|_ | _ _|_ | | _ _|_ | | |
%e A211978 . _ _ _ _ | _ _ | | _ _ | | | _ _ | | | |
%e A211978 . _ _ | _ | | _ | | | _ | | | | _ | | | | |
%e A211978 . | | | | | | | | | | | | | | | | | | | | |
%e A211978 .
%e A211978 . 2 6 12 24 40 70
%e A211978 .
%e A211978 Also using the elements from the diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n) as shown below:
%e A211978 .
%e A211978 11...........................................................
%e A211978 . /\
%e A211978 . / \
%e A211978 . / \
%e A211978 7.................................. / \
%e A211978 . /\ / \
%e A211978 5.................... / \ /\/ \
%e A211978 . /\ / \ /\ / \
%e A211978 3.......... / \ / \ / \/ \
%e A211978 2..... /\ / \ /\/ \ / \
%e A211978 1.. /\ / \ /\/ \ / \ /\/ \
%e A211978 0 /\/ \/ \/ \/ \/ \
%e A211978 . 0,2, 6, 12, 24, 40, 70...
%e A211978 .
%p A211978 Q := sum(x^j/(1-x^j), j = 1 .. i): R := product(1-x^j, j = 1 .. i): g := sum(x^i*(1+i+Q)/R, i = 1 .. 100): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 41); # _Emeric Deutsch_, Oct 07 2016
%t A211978 Array[2 Sum[DivisorSigma[0, m] PartitionsP[# - m], {m, #}] &, 42, 0] (* _Michael De Vlieger_, Mar 20 2020 *)
%Y A211978 Cf. A139582 is twice A000041. A220909 is twice A066186.
%Y A211978 Cf. A006128, A135010, A141285, A186114, A193870, A187219, A194446, A194447, A206437, A211026, A220517, A225600, A278355.
%K A211978 nonn
%O A211978 0,2
%A A211978 _Omar E. Pol_, Jan 03 2013