A331334 -id:A331334 - cp's OEIS frontend

A331333 Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 2, 2, 8, 4, 6, 36, 36, 8, 24, 192, 288, 128, 16, 120, 1200, 2400, 1600, 400, 32, 720, 8640, 21600, 19200, 7200, 1152, 64, 5040, 70560, 211680, 235200, 117600, 28224, 3136, 128, 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256

Offset: 0

Peter Luschny, Jan 19 2020

			Triangle starts:
  [0] 1
  [1] 1,     2
  [2] 2,     8,      4
  [3] 6,     36,     36,      8
  [4] 24,    192,    288,     128,     16
  [5] 120,   1200,   2400,    1600,    400,     32
  [6] 720,   8640,   21600,   19200,   7200,    1152,   64
  [7] 5040,  70560,  211680,  235200,  117600,  28224,  3136,   128
  [8] 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256

T(n, 0) = A000142(n), T(n, n) = A000079(n).
Row sums: A087912, alternating row sums: A295382, antidiagonal sums: A222467, positive half sums: A129683, negative half sums: A331334.
Cf. A021009.

A331333 := proc(n, k) local S; S := proc(n, k) option remember;
`if`(k = 0, 1, `if`(k > n, 0, 2*S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
seq(seq(A331333(n, k), k=0..n), n=0..8);

T(n, k) = n!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else 2*S(n-1, k-1)/k + S(n-1, k).
From Peter Bala, Jan 19 2020: (Start)
T(n,k) = 2^k*(n!/k!)*binomial(n,k).
E.g.f.: exp((2*x*t)/(1 - x))/(1 - x) = 1 + (1 + 2*t)*x + (2 + 8*t + 4*t^2)*x^2/2! + .... Cf. A021009. (End)

cp's OEIS Frontend

A331333 Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.

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