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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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A346839 Decimal expansion of Sum_{n>=0} A346838(n) / n!.

Original entry on oeis.org

2, 9, 3, 4, 0, 7, 9, 9, 3, 0, 2, 6, 0, 2, 3, 3, 8, 7, 4, 0, 4, 7, 7, 8, 4, 3, 3, 9, 4, 0, 2, 9, 0, 0, 2, 0, 3, 8, 3, 1, 4, 5, 8, 8, 2, 7, 1, 2, 4, 8, 9, 2, 6, 0, 4, 5, 1, 2, 1, 1, 6, 7, 2, 1, 5, 9, 9, 5, 4, 0, 1, 4, 1, 0, 9, 8, 6, 9, 4, 1, 6, 3, 6, 6, 7, 4, 1, 4, 4, 8
Offset: 0

Views

Author

Peter Luschny, Aug 13 2021

Keywords

Examples

			0.29340799302602338740477843394029002038314588271248926...

Crossrefs

Cf. A049471, A073448, A346838.

Programs

  • Maple
    A := n -> I*(polylog(-n, -I) - exp(I*Pi*n)*polylog(-n, I)) / exp(I*Pi*n/2):
Digits := 120; sum(A(n)/n!, n = 0..infinity): evalf(%)*10^91:
ListTools:-Reverse(convert(floor(%), base, 10));
  • Mathematica
    RealDigits[Sec[1] - Tan[1], 10 , 100][[1]] (* Amiram Eldar, Aug 17 2021 *)

Formula

Equals sec(1) - tan(1) = A073448 - A049471. - Amiram Eldar, Aug 17 2021

A363393 Triangle read by rows. T(n, k) = [x^k] (2 - Sum_{k=0..n} binomial(n, k)*Euler(k, 1)*(-2*x)^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, -2, 1, 4, 0, -8, 0, 1, 5, 0, -20, 0, 16, 1, 6, 0, -40, 0, 96, 0, 1, 7, 0, -70, 0, 336, 0, -272, 1, 8, 0, -112, 0, 896, 0, -2176, 0, 1, 9, 0, -168, 0, 2016, 0, -9792, 0, 7936, 1, 10, 0, -240, 0, 4032, 0, -32640, 0, 79360, 0
Offset: 0

Views

Author

Peter Luschny, Jun 04 2023

Keywords

Comments

The Swiss-Knife polynomials (A081658 and A153641) generate the dual triangle ('dual' in the sense of Euler-tangent versus Euler-secant numbers).

Examples

			The triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 0;
[3] 1, 3, 0,   -2;
[4] 1, 4, 0,   -8, 0;
[5] 1, 5, 0,  -20, 0,   16;
[6] 1, 6, 0,  -40, 0,   96, 0;
[7] 1, 7, 0,  -70, 0,  336, 0,  -272;
[8] 1, 8, 0, -112, 0,  896, 0, -2176, 0;
[9] 1, 9, 0, -168, 0, 2016, 0, -9792, 0, 7936;

Links

Crossrefs

Cf. A122045 (alternating row sums), A119880 (row sums), A214447 (central column), A155585 (main diagonal), A109573 (subdiagonal), A162660 (variant), A000364.
Cf. A081658, A153641, A220901, A318254, A346838.

Programs

  • Maple
    P := n -> add(binomial(n + 1, j)*bernoulli(j, 1)*(4^j - 2^j)*x^(j-1), j = 0..n+1) / (n + 1):  T := (n, k) -> coeff(P(n), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
# Second program, based on the generating functions of the columns:
ogf := n -> -(-2)^n * euler(n, 1) / (x - 1)^(n + 1):
ser := n -> series(ogf(n), x, 16):
T := (n, k) -> coeff(ser(k), x, n - k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative, based on the bivariate generating function:
egf :=  exp(x*y) * (1 + tanh(y)): ord := 20:
sery := series(egf, y, ord): polx := n -> coeff(sery, y, n):
coefx := n -> seq(n! * coeff(polx(n), x, n - k), k = 0..n):
for n from 0 to 9 do coefx(n) od;
  • Python
    from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k == 0: return 1
    if k % 2 == 0:  return 0
    if k == n: return 1 - sum(T(n, j) for j in range(1, n, 2))
    return (T(n - 1, k) * n) // (n - k)
for n in range(10): print([T(n, k) for k in range(n + 1)])
  • SageMath
    def B(n: int):
    return bernoulli_polynomial(1, n)
def P(n: int):
    return sum(binomial(n + 1, j) * B(j) * (4^j - 2^j) * x^(j - 1)
           for j in range(n + 2)) / (n + 1)
for n in range(10): print(P(n).list())

Formula

For a recursion see the Python program.
T(n, k) = [x^k] P(n, x) where P(n, x) = (1 / (n + 1)) * Sum_{j=0..n+1} binomial(n + 1, j) * Bernoulli(j, 1) * (4^j - 2^j) * x^(j - 1).
Integral_{x=-n..n} P(n, x)/2 dx = n.
T(n, k) = [x^(n - k)] -(-2)^k * Euler(k, 1) / (x - 1)^(k + 1).
T(n, k) = n! * [x^(n - k)][y^n] exp(x*y) * (1 + tanh(y)).

Extensions

Simpler name by Peter Luschny, Nov 17 2024
Showing 1-2 of 2 results.

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