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

%I A331333 #22 Dec 20 2024 03:41:53
%S A331333 1,1,2,2,8,4,6,36,36,8,24,192,288,128,16,120,1200,2400,1600,400,32,
%T A331333 720,8640,21600,19200,7200,1152,64,5040,70560,211680,235200,117600,
%U A331333 28224,3136,128,40320,645120,2257920,3010560,1881600,602112,100352,8192,256
%N A331333 Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.
%F A331333 T(n, k) = n!*S(n, k) where S(n, k) is recursively defined by:
%F A331333 if k = 0 then 1 else if k > n then 0 else 2*S(n-1, k-1)/k + S(n-1, k).
%F A331333 From _Peter Bala_, Jan 19 2020: (Start)
%F A331333 T(n,k) = 2^k*(n!/k!)*binomial(n,k).
%F A331333 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)
%e A331333 Triangle starts:
%e A331333   [0] 1
%e A331333   [1] 1,     2
%e A331333   [2] 2,     8,      4
%e A331333   [3] 6,     36,     36,      8
%e A331333   [4] 24,    192,    288,     128,     16
%e A331333   [5] 120,   1200,   2400,    1600,    400,     32
%e A331333   [6] 720,   8640,   21600,   19200,   7200,    1152,   64
%e A331333   [7] 5040,  70560,  211680,  235200,  117600,  28224,  3136,   128
%e A331333   [8] 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256
%p A331333 A331333 := proc(n, k) local S; S := proc(n, k) option remember;
%p A331333 `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:
%p A331333 seq(seq(A331333(n, k), k=0..n), n=0..8);
%Y A331333 T(n, 0) = A000142(n), T(n, n) = A000079(n).
%Y A331333 Row sums: A087912, alternating row sums: A295382, antidiagonal sums: A222467, positive half sums: A129683, negative half sums: A331334.
%Y A331333 Cf. A021009.
%K A331333 nonn,tabl
%O A331333 0,3
%A A331333 _Peter Luschny_, Jan 19 2020