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Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (j + 1)^n), (tangent case).
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%I #20 Oct 06 2023 04:22:49

%S 1,3,2,7,16,9,15,88,135,64,31,416,1296,1536,625,63,1824,10206,22528,

%T 21875,7776,127,7680,72171,262144,453125,373248,117649,255,31616,

%U 478953,2670592,7265625,10357632,7411887,2097152,511,128512,3057426,25034752,100000000,218350080,265180846,167772160,43046721

%N Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (j + 1)^n), (tangent case).

%C Here we give an inclusion-exclusion representation of 2^n*Euler(n, 1) = A155585(n), in A363398 we give such a representation for 2^n*Euler(n), and in A363400 one for the combined sequences.

%H <a href="/index/Eu#Euler">Index entries for sequences related to Euler numbers.</a>

%F Sum_{k=0..n} (-1)^k * T(n, k) = 2^n*Euler(n, 1) = (-2)^n*Euler(n, 0) = A155585(n).

%F From _Detlef Meya_, Oct 04 2023: (Start)

%F T(n, k) = (k + 1)^n*binomial(n + 1, k + 1)*hypergeom([1, k - n], [k + 2], -1).

%F T(n, k) = (k + 1)^n * (2^(n + 1) - add(binomial(n + 1, j), j=0..k)). (End)

%e The triangle T(n, k) begins:

%e [0] 1;

%e [1] 3, 2;

%e [2] 7, 16, 9;

%e [3] 15, 88, 135, 64;

%e [4] 31, 416, 1296, 1536, 625;

%e [5] 63, 1824, 10206, 22528, 21875, 7776;

%e [6] 127, 7680, 72171, 262144, 453125, 373248, 117649;

%e [7] 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152;

%p P := (n, x) -> add(add(x^j*binomial(k, j)*(j + 1)^n, j=0..k)*2^(n - k), k = 0..n):

%p T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..8);

%t (* From _Detlef Meya_, Oct 04 2023: (Start) *)

%t T[n_, k_] := (k+1)^n*(2^(n+1)-Sum[Binomial[n+1, j], {j, 0, k}]);

%t (* Or *)

%t T[n_, k_] := (k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];

%t Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]] (* End *)

%Y Cf. A155585 (alternating row sums), A363397 (row sums), A126646 (column 0), A000169 (main diagonal), A163395 (central terms), A084623.

%Y Cf. A363398 (secant case), A363400 (combined case).

%K nonn,tabl

%O 0,2

%A _Peter Luschny_, May 31 2023