A363400 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) * ((2 - (n mod 2)) * j + 1)^n).
1, 3, 2, 7, 36, 25, 15, 88, 135, 64, 31, 2106, 10000, 14406, 6561, 63, 1824, 10206, 22528, 21875, 7776, 127, 87480, 1546875, 7529536, 15411789, 14172488, 4826809, 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152
Offset: 0
Examples
Triangle T(n, k) starts: [0] 1; [1] 3, 2; [2] 7, 36, 25; [3] 15, 88, 135, 64; [4] 31, 2106, 10000, 14406, 6561; [5] 63, 1824, 10206, 22528, 21875, 7776; [6] 127, 87480, 1546875, 7529536, 15411789, 14172488, 4826809; [7] 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152;
Crossrefs
Programs
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Maple
P := (n, x) -> add(add(x^j * binomial(k, j) * ((2 - irem(n, 2)) * j + 1)^n, j = 0..k) * 2^(n - k), k = 0..n): T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..8);
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Mathematica
From Detlef Meya, Oct 04 2023: (Start) T[n_, k_] := (2^(n+1)-Binomial[n+1, n-k+1]*Hypergeometric2F1[1, -k, n-k+2, -1])*(2*k+1-k*Mod[n, 2])^n; (* Or: *) T[n_, k_] := (2*k+1-k*Mod[n, 2])^(n-1)*Sum[Binomial[n+1, j], {j, 0, n-k}]*(2*k+1-k*Mod[n, 2]); Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]] (* End *)
Formula
(Sum_{k=0..n} (-1)^k * T(n, k)) / h(n) = A000111(n), where h(n) = (-1)^binomial(n, 2) * 2^(n * iseven(n)), see A059222.
From Detlef Meya, Oct 04 2023: (Start)
T(n, k) = (2*k + 1 - k*(n mod 2))^(n - 1)*add(binomial(n + 1, j), j = 0..n - k)*(2*k + 1 - k*(n mod 2)).
T(n, k) = (2^(n + 1) - binomial(n + 1, n - k + 1)*hypergeom([1, -k], [n - k + 2], -1))*(2*k + 1 - k*(n mod 2))^n. (End)
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