A270917 -id:A270917 - cp's OEIS frontend

A073229 Decimal expansion of e^(1/e).

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

Offset: 1

Rick L. Shepherd, Jul 22 2002

e^(1/e) = 1/((1/e)^(1/e)) (reciprocal of A072364).
Let w(n+1)=A^w(n); then w(n) converges if and only if (1/e)^e <= A <= e^(1/e) (see the comments in A073230) for initial value w(1)=A. If A=e^(1/e) then lim_{n->infinity} w(n) = e. - Benoit Cloitre, Aug 06 2002; corrected by Robert FERREOL, Jun 12 2015
x^(1/x) is maximum for x = e and the maximum value is e^(1/e). This gives an interesting and direct proof that 2 < e < 4 as 2^(1/2) < e^(1/e) > 4^(1/4) while 2^(1/2) = 4^(1/4). - Amarnath Murthy, Nov 26 2002
For large n, A234604(n)/A234604(n-1) converges to e^(1/e). - Richard R. Forberg, Dec 28 2013
Value of the unique base b > 0 for which the exponential curve y=b^x and its inverse y=log_b(x) kiss each other; the kissing point is (e,e). - Stanislav Sykora, May 25 2015
Actually, there is another base with such property, b=(1/e)^e with kiss point (1/e,1/e). - Yuval Paz, Dec 29 2018
The problem of finding the maximum of f(x) = x^(1/x) was posed and solved by the Swiss mathematician Jakob Steiner (1796-1863) in 1850. - Amiram Eldar, Jun 17 2021

			1.44466786100976613365833910859...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 35.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Ovidiu Furdui, Problem 11982, The American Mathematical Monthly, Vol. 124, No. 5 (2017), p. 465; A Limit of a Power of a Sum, Solution to Problem 11982 by Roberto Tauraso, ibid., Vol. 126, No. 2 (2019), p. 187.
Simon Plouffe, exp(1/e).
Jonathan Sondow and Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, Vol. 37 (2010), pp. 151-164; see Definition 4.1 on p. 158.
Jacob Steiner, Über das größte Product der Theile oder Summanden jeder Zahl, Crelle, Vol. 40 (1850), pp. 208; alternative link.
Eric Weisstein's World of Mathematics, Steiner's Problem.

Cf. A001113 (e), A068985 (1/e), A073230 ((1/e)^e), A072364 ((1/e)^(1/e)), A073226 (e^e).
Cf. A093157, A103476.
Cf. A038051, A086331, A252782, A270593, A270917, A270923, A277473, A309652, A332408.

evalf[110](exp(exp(-1))); # Muniru A Asiru, Dec 29 2018

Mathematica
```
RealDigits[ E^(1/E), 10, 110] [[1]]
```
PARI
```
exp(1)^exp(-1)
```

Equals 1 + Integral_{x = 1/e..1} (1 + log(x))/x^x dx = 1 - Integral_{x = 0..1/e} (1 + log(x))/x^x dx. - Peter Bala, Oct 30 2019
Equals Sum_{k>=0} exp(-k)/k!. - Amiram Eldar, Aug 13 2020
Equals lim_{x->oo} (Sum_{n>=1} (x/n)^n)^(1/x) (Furdui, 2017). - Amiram Eldar, Mar 26 2022

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 206, 855, 3585, 15155, 64525, 276278, 1188353, 5130999, 22226049, 96544003, 420368858, 1834203955, 8018057345, 35107961175, 153950675585, 675978772326, 2971700764941, 13078268135683, 57613905606273, 254038914924791, 1121081799217231

Offset: 0

Vaclav Kotesovec, Mar 25 2016

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),
       (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 31 2021

Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[(-1)^j*x^j/(j*(x^j - 1)), {j, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 26 2019

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...

A252782 a(n) = n-th term of Euler transform of n-th powers.

Original entry on oeis.org

1, 1, 5, 36, 490, 12729, 689896, 70223666, 13803604854, 5567490203192, 4386006155453382, 6711625359213752077, 21048250447828058144403, 131214686495783317936950378, 1603891839732647136012816743764, 40296598014204065945778862754895836

Offset: 0

Alois P. Heinz, Dec 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..80
Vaclav Kotesovec, Graph - The asymptotic ratio

Main diagonal of A144048.

Cf. A008485, A073229, A255672, A270917.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
       d*d^k, d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..20);

Table[SeriesCoefficient[Product[1/(1-x^k)^(k^n),{k,1,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 01 2015 *)

a(n) = [x^n] Product_{j>=1} 1/(1-x^j)^(j^n).

Conjecture: limit n->infinity a(n)^(1/n^2) = exp(exp(-1)) = 1.444667861... . - Vaclav Kotesovec, Mar 25 2016

A270922 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k*n).

Original entry on oeis.org

1, 1, 5, 28, 141, 751, 4064, 22198, 122381, 679375, 3792155, 21263331, 119679000, 675763232, 3826165838, 21715370653, 123502583565, 703694143160, 4016079632039, 22953901314649, 131366012754691, 752709483123304, 4317601694413683, 24790635783551008

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A255672, A270913, A270917, A270924, A380291.

Table[SeriesCoefficient[Product[(1+x^k)^(k*n), {k, 1, n}], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = 5.86811560195778704624328861800917668... and c = 0.25351514412215050116013727161633502...

a(n) = [x^n] exp(n*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A284992 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j^k) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 8, 3, 1, 1, 16, 35, 31, 16, 4, 1, 1, 32, 97, 119, 83, 28, 5, 1, 1, 64, 275, 457, 433, 201, 49, 6, 1, 1, 128, 793, 1763, 2297, 1476, 487, 83, 8, 1, 1, 256, 2315, 6841, 12421, 11113, 4962, 1141, 142, 10, 1, 1

Offset: 0

Seiichi Manyama, Apr 07 2017

			Square array begins:
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  2,   4,    8,    16,     32,      64,      128, ...
  2,  5,  13,   35,    97,    275,     793,     2315, ...
  2,  8,  31,  119,   457,   1763,    6841,    26699, ...
  3, 16,  83,  433,  2297,  12421,   68393,   382573, ...
  4, 28, 201, 1476, 11113,  85808,  678101,  5466916, ...
  5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give A000009, A026007, A027998, A248882, A248883, A248884.
Rows (0+1),2-3 give: A000012, A000079, A007689.
Main diagonal gives A270917.
Cf. A283272, A284993.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 16 2017

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} (1+x^j)^(j^k).

A270923 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^(k^n).

Original entry on oeis.org

1, 2, 10, 88, 1414, 46648, 3026028, 373615284, 92794268694, 46265940243794, 44694344296430280, 86689242777435107120, 340600515192402995860548, 2624923513793602103874986688, 40749869155795866122979193705136, 1290021269710020392957588463834452744

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..80

Cf. A252782, A270917, A270919, A270924.

Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

Conjecture: limit n->infinity a(n)^(1/n^2) = exp(exp(-1)) = 1.444667861...

a(n) = [x^n] exp(Sum_{k>=1} (sigma_(n+1)(2*k) - sigma_(n+1)(k))*x^k/(2^n*k)). - Ilya Gutkovskiy, Apr 26 2019

A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

Original entry on oeis.org

1, -1, -3, -1, 25, 624, 9871, 170470, 3027249, 55077245, 979330606, 15079702923, 94670678245, -7958168036625, -626145997536240, -34564907982551791, -1733699815491494303, -84294315853736719077, -4067859614343931897505, -196552300464314521511610, -9519733465269825759734169

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			The table of coefficients of x^k in expansion of Product_{k>=1} (1 - x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),  -1,    0,   0,     1,  ...
n = 2:  1,  -2,  (-3),   0,   2,    12,  ...
n = 3:  1,  -3,   -6,  (-1),  9,    63,  ...
n = 4:  1,  -4,  -10,   -4, (25),  224,  ...
n = 5:  1,  -5,  -15,  -10,  55,  (624), ...

Cf. A010815, A008705, A252654, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300458.

Table[SeriesCoefficient[Product[(1 - x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

Original entry on oeis.org

1, -1, -1, -10, 11, 374, 9792, 183847, 3469427, 65038049, 1195396233, 19667738452, 189089161562, -6219720781782, -606316892131934, -35104997710496175, -1795953382595105853, -88223902016631657740, -4283800987347611165184, -207864171877269042498096, -10102590396625592962089500

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			The table of coefficients of x^k in expansion of Product_{k>=1} 1/(1 + x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),   0,   -1,   1,    -1,  ...
n = 2:  1,  -2,  (-1),  -4,   3,    -2,  ...
n = 3:  1,  -3,   -3, (-10),  6,    15,  ...
n = 4:  1,  -4,   -6,  -20, (11),  104,  ...
n = 5:  1,  -5,  -10,  -35,  20,  (374), ...

Cf. A081362, A252654, A255526, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300457.

Table[SeriesCoefficient[Product[1/(1 + x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

cp's OEIS Frontend

A073229 Decimal expansion of e^(1/e).

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A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

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A252782 a(n) = n-th term of Euler transform of n-th powers.

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A270922 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k*n).

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A284992 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j^k) in powers of x.

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A270923 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^(k^n).

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A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

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A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

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