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.
0, 2, 6, 13, 26, 51, 98, 182, 324, 552, 902, 1419, 2158, 3185, 4578, 6428, 8840, 11934, 15846, 20729, 26754, 34111, 43010, 53682, 66380, 81380, 98982, 119511, 143318, 170781, 202306, 238328, 279312, 325754, 378182, 437157, 503274, 577163, 659490, 750958
Offset: 1
Examples
a(1)=0 since we can't choose 2 nonempty blocks/cells (i.e., subintervals) from an interval of one block. 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). 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: 6|0 (second interval empty): C(6,2) = 15 from the first interval; 0|6 (first interval empty): C(6,2) = 15 from the second interval; 2|4: C(2,2)*C(4,2) = 6; 3|3: C(3,2)*C(3,2) = 9; 4|2: C(4,2)*C(2,2) = 6.
Links
- Paolo Xausa, Table of n, a(n) for n = 1..10000
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Programs
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Mathematica
A353232[n_] := 2*Binomial[n, 2] + Binomial[n + 1, 5]; Array[A353232, 50] (* Paolo Xausa, May 27 2024 *)
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PARI
a(n) = 2*binomial(n,2) + binomial(n+1,5); \\ Michel Marcus, Jul 06 2022
Formula
a(n) = 2*C(n,2) + C(n+1,5).
G.f.: x^2*(2 - 6*x + 7*x^2 - 2*x^3)/(1 - x)^6. - Stefano Spezia, May 02 2022
a(n) = n*(n-1)*(n^3 - 4*n^2 + n + 126)/120. - R. J. Mathar, Jul 05 2022
Comments