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A376878 Triangle read by rows: T(n, k) = n^k * n! * [x^k][y^n]((sec(y) + tan(y)) * exp(x*y)).

%I A376878 #15 Apr 14 2025 05:34:51
%S A376878 1,1,1,1,4,4,2,9,27,27,5,32,96,256,256,16,125,500,1250,3125,3125,61,
%T A376878 576,2700,8640,19440,46656,46656,272,2989,16464,60025,168070,352947,
%U A376878 823543,823543,1385,17408,109312,458752,1433600,3670016,7340032,16777216,16777216
%N A376878 Triangle read by rows: T(n, k) = n^k * n! * [x^k][y^n]((sec(y) + tan(y)) * exp(x*y)).
%F A376878 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 A376878 T(n, k) = n^k*A109449(n, k) = n^k*binomial(n, k)*A000111(n - k).
%e A376878 Triangle starts:
%e A376878   [0]    1;
%e A376878   [1]    1,     1;
%e A376878   [2]    1,     4,      4;
%e A376878   [3]    2,     9,     27,     27;
%e A376878   [4]    5,    32,     96,    256,     256;
%e A376878   [5]   16,   125,    500,   1250,    3125,    3125;
%e A376878   [6]   61,   576,   2700,   8640,   19440,   46656,   46656;
%e A376878   [7]  272,  2989,  16464,  60025,  168070,  352947,  823543,   823543;
%e A376878   [8] 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216;
%p A376878 P := n -> coeff(series((sec(y) + tan(y)) * exp(x*y), y, 12), y, n):
%p A376878 seq(seq(coeff(P(n), x,  k) * n^k * n!, k = 0..n), n = 0..8);
%p A376878 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 A376878 seq(print([n], seq(T(n, k), k = 0..n)), n = 0..8);
%o A376878 (Python)
%o A376878 from math import comb, isqrt
%o A376878 from sympy import bernoulli, euler
%o A376878 def A000111(n): return abs(((1<<n+1)-1<<n+1)*bernoulli(n+1)//(n+1) if n&1 else euler(n))
%o A376878 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 A376878 Cf. A000111, A000312, A079901, A109449, A292976 (row sums).
%K A376878 nonn,tabl
%O A376878 0,5
%A A376878 _Peter Luschny_, Oct 13 2024