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A363400
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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) * ((2 - (n mod 2)) * j + 1)^n).
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3
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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
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OFFSET
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0,2
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COMMENTS
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In A363398 we give an inclusion-exclusion representation for 2^n*Euler(n), and in A363399 we give such a representation of 2^n*Euler(n, 1) = A155585(n). Here the two representations are combined into one of A000111.
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LINKS
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FORMULA
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T(n, k) = A363399(n, k) for 0 <= k <= n if n is odd otherwise A363398(n, k).
(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.
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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EXAMPLE
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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;
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MAPLE
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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
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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