A215077 -id:A215077 - cp's OEIS frontend

A202357 Decimal expansion of the number x satisfying e*x = e^(-x).

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

Offset: 0

Clark Kimberling, Dec 18 2011

See A202322 for a guide to related sequences. The Mathematica program includes a graph.

			0.2784645427610737951093587390229801554394774886...

Heine Halberstam and Hans Egon-Richert, Sieve Methods, Dover Publications (2011). See Theorem 2.1.

G. C. Greubel, Table of n, a(n) for n = 0..5000
W. Gautschi, The incomplete Gamma Function since Tricomi, Atti Conv. Lincei 147 (1999) 203, eq. (2.16).
H. J. H. Tuenter, On the generalized Poisson distribution, arXiv:math/0606238 [math.ST], 2006. Published version on On the Generalized Poisson Distribution, Statistica Neerlandica, 54(3):374-376, November 2000.
S. M. Zemyan, On the zeros of the Nth partial sum of the exponential series, Am. Math. Monthly 112 (2005) 891-909.
Index entries for transcendental numbers.

Cf. A202322, A215077, A218300.

u = E; v = 0;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .27, .28}, WorkingPrecision -> 110]
RealDigits[r]  (* A202357 *)
RealDigits[ ProductLog[1/E], 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[LambertW[Exp[-1]],10,120][[1]] (* Harvey P. Dale, Dec 24 2019 *)

lambertw(exp(-1)) \\ Michel Marcus, Mar 21 2016

The constant in A202355 minus 1. - R. J. Mathar, Dec 21 2011
1+x+log(x)=0. - R. J. Mathar, Nov 02 2012
Equals LambertW(exp(-1)). - Vaclav Kotesovec, Jan 10 2014

A215078 Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 2, 5, 0, 3, 27, 36, 0, 4, 102, 392, 354, 0, 5, 330, 2760, 6500, 4425, 0, 6, 975, 15880, 73350, 123090, 67171, 0, 7, 2709, 81060, 654500, 2033325, 2637327, 1200304, 0, 8, 7196, 381808, 5064780, 25926824, 59992660, 63259168, 24684612, 0, 9, 18468, 1696464, 35574840, 281668590, 1034305524, 1896003648, 1681960464, 574304985, 0, 10, 46125, 7208880, 232816500, 2740317300, 14981494710, 42457884000, 64240088580, 49143419250, 14914341925

Offset: 0

Olivier Gérard, Aug 02 2012

If one starts the sum at j=0, the initial term T(0,0) is 1.

			  0
  0  1
  0  2    5
  0  3   27       36
  0  4  102      392      354
  0  5  330     2760     6500     4425
  0  6  975    15880    73350   123090    67171
  0  7 2709    81060   654500  2033325  2637327  1200304

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Binomial convolution of A215083.

Cf. A215077 (row sums), A031971 (diagonal)

A215078 := proc(n,k)
        binomial(n,k)*add(j^n,j=1..k) ;
end proc:
seq(seq(A215078(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jan 27 2023

Flatten[Table[Table[Sum[j^n, {j, 1, k}]*Binomial[n, k], {k, 0, n}], {n, 0, 10}], 1]

T(n,k) = binomial(n,k)*sum(j^n, j=1..k)

A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).

Original entry on oeis.org

1, 0, 1, 0, 3, 4, 0, 7, 32, 27, 0, 15, 176, 405, 256, 0, 31, 832, 3888, 6144, 3125, 0, 63, 3648, 30618, 90112, 109375, 46656, 0, 127, 15360, 216513, 1048576, 2265625, 2239488, 823543, 0, 255, 63232, 1436859, 10682368, 36328125, 62145792, 51883209, 16777216, 0, 511, 257024, 9172278, 100139008, 500000000, 1310100480, 1856265922, 1342177280, 387420489, 0, 1023, 1037312, 57159432, 889192448, 6230468750, 23339943936, 49715643824, 60129542144, 38354628411, 10000000000

Offset: 0

Olivier Gérard, Aug 02 2012

Initial term T(0,0) may be computed as 0, depending on formula and convention.

			      1
      0       1
      0       3       4
      0       7      32      27
      0      15     176     405     256
      0      31     832    3888    6144    3125
      0      63    3648   30618   90112  109375   46656
      0     127   15360  216513 1048576 2265625 2239488  823543

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Row sums sequence is A215077.
Product of A055248 and A089072 (with an initial 0 in each row).
Cf. A000225 (column k=1), A000312 (diagonal).

A215079 := proc(n,k)
    k^n*add( binomial(n,n-k-j),j=0..n-k) ;
end proc: # R. J. Mathar, Feb 08 2021

Flatten[Table[Table[Sum[k^n*Binomial[n, n - k - j], {j, 0, n - k}],  {k, 0, n}], {n, 0, 10}], 1]

T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k) = k^n * A055248(n,k-1).
T(n,k) = k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1)
T(n,1) = A000225(n). - R. J. Mathar, Feb 08 2021

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A202357 Decimal expansion of the number x satisfying e*x = e^(-x).

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A215078 Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.

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Mathematica

Formula

A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).

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