A376878 - OEIS

%I #11 Nov 13 2024 01:54:11

%S 1,1,1,1,4,4,2,9,27,27,5,32,96,256,256,16,125,500,1250,3125,3125,61,

%T 576,2700,8640,19440,46656,46656,272,2989,16464,60025,168070,352947,

%U 823543,823543,1385,17408,109312,458752,1433600,3670016,7340032,16777216,16777216

%N Triangle read by rows: T(n, k) = n^k * n! * [x^k][y^n]((sec(y) + tan(y)) * exp(x*y)).

%F T(n, k) = (-1)^binomial(n-k, 2)*n^k*binomial(n, k)*(Euler(n-k) - Euler(n-k, 0)*2^(n - k))) for 0 <= k < n and n^n for n = k.

%F T(n, k) = n^k*A109449(n, k) = n^k*binomial(n, k)*A000111(n - k).

%e Triangle starts:

%e   [0]    1;

%e   [1]    1,     1;

%e   [2]    1,     4,      4;

%e   [3]    2,     9,     27,     27;

%e   [4]    5,    32,     96,    256,     256;

%e   [5]   16,   125,    500,   1250,    3125,    3125;

%e   [6]   61,   576,   2700,   8640,   19440,   46656,   46656;

%e   [7]  272,  2989,  16464,  60025,  168070,  352947,  823543,   823543;

%e   [8] 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216;

%p P := n -> coeff(series((sec(y) + tan(y)) * exp(x*y), y, 12), y, n):

%p seq(seq(coeff(P(n), x,  k) * n^k * n!, k = 0..n), n = 0..8);

%p T := (n, k) -> ifelse(n = k, n^n, (-1)^binomial(n - k, 2)*n^k*binomial(n, k)*(euler(n - k) - euler(n - k, 0)*2^(n - k))):

%p seq(print([n], seq(T(n, k), k = 0..n)), n = 0..8);

%o (Python)

%o from math import comb, isqrt

%o from sympy import bernoulli, euler

%o def A000111(n): return abs(((1<<n+1)-1<<n+1)*bernoulli(n+1)//(n+1) if n&1 else euler(n))

%o def A376878(n): return comb(a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),b:=n-comb(a+1,2))*a**b*A000111(a-b) # _Chai Wah Wu_, Nov 13 2024

%Y Cf. A000111, A000312, A079901, A109449, A292976 (row sums).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Oct 13 2024