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.

Showing 1-3 of 3 results.

A363397 a(n) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} binomial(k, j) * (j + 1)^n. Row sums of A363399.

Original entry on oeis.org

1, 5, 32, 302, 3904, 64272, 1286144, 30313712, 822571008, 25258008320, 865863532544, 32779942009344, 1358320701014016, 61149815860711424, 2971951570679234560, 155090406558662064128, 8649258967534890123264, 513370937392454603833344
Offset: 0

Views

Author

Peter Luschny, Jun 02 2023

Keywords

Crossrefs

Cf. A363399.

Programs

  • Maple
    a := n -> add(add(binomial(k, j)*(j + 1)^n, j=0..k)*2^(n - k), k = 0..n):
seq(a(n), n = 0..17);
  • Mathematica
    Table[Sum[2^(n-k) * Sum[Binomial[k, j] * (j+1)^n, {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 02 2023 *)

Formula

a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^n / ((1 - LambertW(exp(-1))) * exp(n) * LambertW(exp(-1))^(n+1)). - Vaclav Kotesovec, Jun 02 2023

A363398 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*j + 1)^n), (secant case).

Original entry on oeis.org

1, 3, 3, 7, 36, 25, 15, 297, 625, 343, 31, 2106, 10000, 14406, 6561, 63, 13851, 131250, 369754, 413343, 161051, 127, 87480, 1546875, 7529536, 15411789, 14172488, 4826809, 255, 540189, 17109375, 134237509, 444816117, 721025327, 564736653, 170859375
Offset: 0

Views

Author

Peter Luschny, May 31 2023

Keywords

Comments

Here we give an inclusion-exclusion representation of 2^n*Euler(n) (see A122045 and A002436), in A363399 we give such a representation for 2^n*Euler(n, 1) = A155585(n), and in A363400 one for the combined sequences.

Examples

			The triangle T(n, k) starts:
  [0]   1;
  [1]   3,      3;
  [2]   7,     36,       25;
  [3]  15,    297,      625,       343;
  [4]  31,   2106,    10000,     14406,      6561;
  [5]  63,  13851,   131250,    369754,    413343,    161051;
  [6] 127,  87480,  1546875,   7529536,  15411789,  14172488,   4826809;
  [7] 255, 540189, 17109375, 134237509, 444816117, 721025327, 564736653, 170859375;

Links

Crossrefs

Cf. A122045 (alternating row sums), A363396 (row sums), A126646 (column 0), A085527 (main diagonal), A141475 (central terms).
Cf. A363399 (tangent case), A363400 (combined case).
Cf. A122045, A002436.

Programs

  • Maple
    P := (n, x) -> add(add(x^j*binomial(k, j)*(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..7);
  • Mathematica
    (* From Detlef Meya, Oct 04 2023: (Start) *)
T[n_, k_] := (2*k+1)^n*(2^(n+1) - Sum[Binomial[n+1, j], {j,0,k}]);
(* Or: *)
T[n_, k_] := (2*k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)

Formula

Sum_{k=0..n} (-1)^k*T(n, k) = 2^n*Euler(n) = 4^n*Euler(n, 1/2).
(Sum_{k=0..n} (-1)^k*T(n, k)) / 2^n = Euler(n) = 2^n*Euler(n, 1/2) = A122045(n).
Sum_{k=0..2*n} (-1)^k*T(2*n, k) = 4^n*Euler(2*n) = 16^n*Euler(2*n, 1/2) = (-1)^n*A002436(n).
From Detlef Meya, Oct 04 2023: (Start)
T(n, k) = (2*k + 1)^n * binomial(n+1, k+1) * hypergeom([1, k-n], [k+2], -1).
T(n, k) = (2*k + 1)^n * (2^(n + 1) - Sum_{j=0..k} binomial(n+1, j)). (End)

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

Views

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;

Links

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)
Showing 1-3 of 3 results.

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