cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

Original entry on oeis.org

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

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Author

Peter Luschny, May 31 2023

Keywords

Comments

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.

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

Cf. A126646 (column 0), A363401 (row sums), A000111, A059222, A002436.
Cf. A363398 (secant case), A363399 (tangent case).

Programs

  • 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);
  • 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

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.
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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