A099554 -id:A099554 - cp's OEIS frontend

A126583 Decimal expansion of solution to exp(-x) = x^2.

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

Offset: 0

Denton J. Dailey (djd1497(AT)aol.com), Jan 05 2007

The value of the infinite power tower function x^x^x... at x = sqrt(1/e). - Alois P. Heinz, Oct 19 2016

			0.7034674224983916520498186018599021303429284310342236...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration
Index entries for transcendental numbers

Cf. A030178, A099554, A202356.

RealDigits[ FindRoot[ Exp[ -x] == x^2, {x, {.5, 1}}, WorkingPrecision -> 120][[1, 2, 1]], 10, 111][[1]]
RealDigits[ 2*ProductLog[1/2], 10, 102] // First (* Jean-François Alcover, Feb 27 2013 *)

2*lambertw(1/2) \\ G. C. Greubel, Mar 06 2018

Equals 2*LambertW(1/2). - Alois P. Heinz, Oct 19 2016

Equals log(A099554) = 2*A202356. - Hugo Pfoertner, Jul 19 2024

A099555 Triangle, read by rows, where T(n,k) = (n-floor(k/2))^k for k = 0..2*n - 1, with T(0,0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 8, 1, 1, 1, 4, 9, 27, 16, 32, 1, 1, 1, 5, 16, 64, 81, 243, 64, 128, 1, 1, 1, 6, 25, 125, 256, 1024, 729, 2187, 256, 512, 1, 1, 1, 7, 36, 216, 625, 3125, 4096, 16384, 6561, 19683, 1024, 2048, 1, 1, 1, 8, 49, 343, 1296, 7776, 15625, 78125, 65536

Offset: 0

Paul D. Hanna, Oct 22 2004

Row functions in y are given by: R_n(y) = Sum_{k=0..2n} (n-floor(k/2))^k*y^k/k!. Evaluated at y=1, the asymptotic behavior of the rows is given by: R_n(1) ~ c*r^n where c = (r+sqrt(r))/(1+2*sqrt(r)) = 0.8957126... and r = 2.0207473586... satisfies r = exp(1/sqrt(r)) -- see A099554 for the decimal expansion of this constant.

			The asymptotic behavior can be demonstrated at the 4th row function:
R_4(y) = 1 + 4*y + 9*y^2/2! + 27*y^3/3! + 16*y^4/4! + 32*y^5/5! + y^6/6! + y^7/7!;
R_4(1) = 14.93492... = (0.895684...)*r^4, where r = 2.0207473586...
Rows begin:
  [1]
  [1, 1],
  [1, 2,  1,   1],
  [1, 3,  4,   8,   1,    1],
  [1, 4,  9,  27,  16,   32,    1,     1],
  [1, 5, 16,  64,  81,  243,   64,   128,    1,     1],
  [1, 6, 25, 125, 256, 1024,  729,  2187,  256,   512,    1,    1],
  [1, 7, 36, 216, 625, 3125, 4096, 16384, 6561, 19683, 1024, 2048, 1, 1],
  ...
which can be derived from the square array A003992:
  [1, 0,  0,   0,   0,    0,     0, ...],
  [1, 1,  1,   1,   1,    1,     1, ...],
  [1, 2,  4,   8,  16,   32,    64, ...],
  [1, 3,  9,  27,  81,  243,   729, ...],
  [1, 4, 16,  64, 256, 1024,  4096, ...],
  [1, 5, 25, 125, 625, 3125, 15625, ...],
  ...
by shifting each column k down by floor(k/2) rows, and omitting the zeros coming from row 0 of A003992.

Cf. A003992, A099554, A099556.

seq(print(`if`(n=0, 1, seq((n - floor(k/2))^k, k=0..2*n-1))), n=0..10); # Georg Fischer, Nov 21 2024

PARI
```
T(n,k)=(n-k\2)^k
```

E.g.f.: ((1-x*cosh(sqrt(x)*y)) + sqrt(x)*sinh(sqrt(x)*y))/(1+x^2-2*x*cosh(sqrt(x)*y)).

Definition corrected by Georg Fischer, Nov 21 2024

cp's OEIS Frontend

A126583 Decimal expansion of solution to exp(-x) = x^2.

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