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 A353232 #50 May 27 2024 15:51:51
%S A353232 0,2,6,13,26,51,98,182,324,552,902,1419,2158,3185,4578,6428,8840,
%T A353232 11934,15846,20729,26754,34111,43010,53682,66380,81380,98982,119511,
%U A353232 143318,170781,202306,238328,279312,325754,378182,437157,503274,577163,659490,750958
%N A353232 a(n) is the number of ways to split [n] = {1,2,...,n} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n} and then, if both intervals are nonempty, select 2 nonempty blocks/cells (i.e., subintervals) from each of them, or if one of the intervals is empty, select 2 nonempty blocks/cells from the nonempty interval.
%C A353232 See A095263 for the number of ways to split [n] into an unspecified number of intervals and then choose 2 blocks (i.e., subintervals) from each interval.
%H A353232 Paolo Xausa, <a href="/A353232/b353232.txt">Table of n, a(n) for n = 1..10000</a>
%H A353232 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A353232 a(n) = 2*C(n,2) + C(n+1,5).
%F A353232 G.f.: x^2*(2 - 6*x + 7*x^2 - 2*x^3)/(1 - x)^6. - _Stefano Spezia_, May 02 2022
%F A353232 a(n) = n*(n-1)*(n^3 - 4*n^2 + n + 126)/120. - _R. J. Mathar_, Jul 05 2022
%e A353232 a(1)=0 since we can't choose 2 nonempty blocks/cells (i.e., subintervals) from an interval of one block.
%e A353232 a(2)=2 since we have 2 cases: first interval is empty, so we choose both blocks (i.e., subintervals) from the second interval in C(2,2) ways, and similarly for the case of the second interval being empty (note we can't consider the case where [2] splits into 2 intervals of one block each since we can't choose 2 nonempty blocks from a single block; i.e., C(1,2)*C(1,2)=0).
%e A353232 a(6)=51 since the following are the number of ways to split [6] into 2 intervals with k and (n-k) blocks (subintervals) each (written as k|(n-k) below) and to choose the blocks/cells:
%e A353232 6|0 (second interval empty): C(6,2) = 15 from the first interval;
%e A353232 0|6 (first interval empty): C(6,2) = 15 from the second interval;
%e A353232 2|4: C(2,2)*C(4,2) = 6;
%e A353232 3|3: C(3,2)*C(3,2) = 9;
%e A353232 4|2: C(4,2)*C(2,2) = 6.
%t A353232 A353232[n_] := 2*Binomial[n, 2] + Binomial[n + 1, 5];
%t A353232 Array[A353232, 50] (* _Paolo Xausa_, May 27 2024 *)
%o A353232 (PARI) a(n) = 2*binomial(n,2) + binomial(n+1,5); \\ _Michel Marcus_, Jul 06 2022
%Y A353232 Cf. A095263, A002378, A000389.
%K A353232 nonn,easy
%O A353232 1,2
%A A353232 _Enrique Navarrete_, May 01 2022