A212852 Number of n X 5 arrays with rows being permutations of 0..4 and no column j greater than column j-1 in all rows.
1, 3651, 966751, 158408751, 21855093751, 2801736968751, 347190069843751, 42328368099218751, 5119530150996093751, 616756797369980468751, 74155772004699902343751, 8907394925520999511718751
Offset: 1
Keywords
Examples
Some solutions for n=3 ..0..3..1..2..4....0..2..4..1..3....0..1..4..3..2....0..2..3..4..1 ..1..0..4..3..2....1..0..3..2..4....1..3..0..4..2....0..4..3..1..2 ..2..4..1..3..0....1..2..0..4..3....3..1..4..0..2....4..0..1..3..2
Links
- R. H. Hardin, Table of n, a(n) for n = 1..210
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
- Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6), p. 248 (with t=0).
- Wikipedia, Multinomial coefficients.
- Wikipedia, Partition (number theory).
Crossrefs
Programs
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Mathematica
T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k - j], {j, 1, k}]]; a[n_] := T[n, 5]; Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz in A212855 *)
Formula
Empirical: a(n) = 246*a(n-1) -20545*a(n-2) +751800*a(n-3) -12911500*a(n-4) +100380000*a(n-5) -304200000*a(n-6) +216000000*a(n-7).
Empirical: a(n) = -2*5^n + 3*20^n - 4*60^n + 120^n + 3*30^n - 2*10^n + 1. R. J. Mathar, Jun 25 2012
Sum_{s = 0..7} (-1)^s * A325305(5, s) * a(n-s) = 0 for n >= 8. (This is the same as R. H. Hardin's recurrence above, and it follows from Eq. (6) (with t=0), p. 248, in Abramson and Promislow (1978).) - Petros Hadjicostas, Sep 06 2019
Comments