A333446 Table T(n,k) read by upward antidiagonals. T(n,k) = Sum_{i=1..n} Product_{j=1..k} (i-1)*k+j.

1, 3, 2, 6, 14, 6, 10, 44, 126, 24, 15, 100, 630, 1704, 120, 21, 190, 1950, 13584, 30360, 720, 28, 322, 4680, 57264, 390720, 666000, 5040, 36, 504, 9576, 173544, 2251200, 14032080, 17302320, 40320, 45, 744, 17556, 428568, 8626800, 110941200, 603353520, 518958720, 362880

Offset

1,2

Comments

T(n,k) is the maximum value of Sum_{i=1..n} Product_{j=1..k} r[(i-1)*k+j] among all permutations r of {1..kn}. For the minimum value see A331889.

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Formula

T(n,k) = Sum_{i=1..n} Gamma(ik+1)/Gamma((i-1)k+1).

Example

From Seiichi Manyama, Jul 23 2020: (Start)
T(3,2) = Sum_{i=1..3} Product_{j=1..2} (i-1)*2+j = 1*2 + 3*4 + 5*6 = 44.
Square array begins:
   1,   2,    6,     24,      120,        720, ...
   3,  14,  126,   1704,    30360,     666000, ...
   6,  44,  630,  13584,   390720,   14032080, ...
  10, 100, 1950,  57264,  2251200,  110941200, ...
  15, 190, 4680, 173544,  8626800,  538459200, ...
  21, 322, 9576, 428568, 25727520, 1940869440, ... (End)

Prog

(Python)
def T(n, k): # T(n, k) for A333446
    c, l = 0, list(range(1, k*n+1, k))
    lt = list(l)
    for i in range(n):
        for j in range(1, k):
            lt[i] *= l[i]+j
        c += lt[i]
    return c

Crossrefs

Column k=1-3 give A000217, A268684, A268685(n-1).

Main diagonal gives A336513.

Cf. A323663, A331889,

Sequence in context: A078091 A073883 A248982 * A289069 A074718 A285457

Adjacent sequences: A333443 A333444 A333445 * A333447 A333448 A333449

Keyword

nonn,tabl

Author

Chai Wah Wu, Mar 23 2020

Status

approved