A086331 -id:A086331 - cp's OEIS frontend

A294353 Product of first n terms of the binomial transform of n^n (A086331).

1, 2, 14, 602, 236586, 1116922506, 78020387811618, 95634036502805444826, 2378081951650318040462277306, 1361239109900199746154166909875717978, 20062823024247092576000017563809908231829439138, 8420023655209092490508999978430595224656730339006712229850

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Cf. A002109, A047656, A086331, A086619, A294352.

Table[Product[1 + Sum[Binomial[m, k]*k^k, {k, 1, m}], {m, 0, n}], {n, 0, 12}]

a(n) ~ c * n^(n*(n+1)/2 + 1/12 + exp(-1)/2) / exp(n^2/4 - n*exp(-1)), where c = 1.981849007720509372587479129359338230641860983165241915197508762536...

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

Original entry on oeis.org

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

A088957 Hyperbinomial transform of the sequence of 1's.

Original entry on oeis.org

1, 2, 6, 29, 212, 2117, 26830, 412015, 7433032, 154076201, 3608522954, 94238893883, 2715385121740, 85574061070045, 2928110179818478, 108110945014584623, 4284188833355367440, 181370804507130015569, 8169524599872649117330, 390114757072969964280163

Offset: 0

Paul D. Hanna, Oct 26 2003

nonn

See A088956 for the definition of the hyperbinomial transform.
a(n) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1.  The triangle A088956 classifies these functions according to the number of undefined elements in the domain.  The triangle A144289 classifies these functions according to the number of edges in their digraph representation (considering the empty function to have 1 edge).  The triangle A203092 classifies these functions according to the number of connected components. - Geoffrey Critzer, Dec 29 2011
a(n) is the number of rooted subtrees (for a fixed root) in the complete graph on n+1 vertices: a(3) = 29 is the number of rooted subtrees in K_4: 1 of size 1, 3 of size 2, 9 of size 3, and 16 spanning subtrees. - Alex Chin, Jul 25 2013 [corrected by Marko Riedel, Mar 31 2019]
From Gus Wiseman, Jan 28 2024: (Start)
Also the number of labeled loop-graphs on n vertices such that it is possible to choose a different vertex from each edge in exactly one way. For example, the a(3) = 29 uniquely choosable loop-graphs (loops shown as singletons) are:
  {}  {1}  {1,2}  {1,12}   {1,2,13}  {1,12,13}
      {2}  {1,3}  {1,13}   {1,2,23}  {1,12,23}
      {3}  {2,3}  {2,12}   {1,3,12}  {1,13,23}
                  {2,23}   {1,3,23}  {2,12,13}
                  {3,13}   {2,3,12}  {2,12,23}
                  {3,23}   {2,3,13}  {2,13,23}
                  {1,2,3}            {3,12,13}
                                     {3,12,23}
                                     {3,13,23}
(End)

			a(5) = 2117 = 1296 + 625 + 160 + 30 + 5 + 1 = sum of row 5 of triangle A088956.

Alois P. Heinz, Table of n, a(n) for n = 0..387
Marko Riedel et al., Proof of e.g.f. of sequence.

Cf. A088956 (triangle).
Row sums of A144289. - Alois P. Heinz, Jun 01 2009
Cf. A086331, A000169.
Column k=1 of A144303. - Alois P. Heinz, Oct 30 2012
The covering case is A000272, also the case of exactly n edges.
Without the choice condition we have A006125 (shifted left).
The unlabeled version is A087803.
The choosable version is A368927, covering A369140, loopless A133686.
The non-choosable version is A369141, covering A369142, loopless A367867.
Cf. A000081, A000085, A057500, A062740, A137916, A277473, A322661, A367904, A368596, A368597, A368924.

a088957 = sum . a088956_row  -- Reinhard Zumkeller, Jul 07 2013

a:= n-> add((n-j+1)^(n-j-1)*binomial(n,j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

nn = 16; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[Exp[x] Exp[t], {x, 0, nn}], x]  (* Geoffrey Critzer, Dec 29 2011 *)
With[{nmax = 50}, CoefficientList[Series[-LambertW[-x]*Exp[x]/x, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)

x='x+O('x^10); Vec(serlaplace(-lambertw(-x)*exp(x)/x)) \\ G. C. Greubel, Nov 14 2017

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k).
E.g.f.: A(x) = exp(x+sum(n>=1, n^(n-1)*x^n/n!)).
E.g.f.: -LambertW(-x)*exp(x)/x. - Vladeta Jovovic, Oct 27 2003
a(n) ~ exp(1+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013
Binomial transform of A000272. - Gus Wiseman, Jan 25 2024

A069856 E.g.f.: exp(x)/(1+LambertW(x)).

Original entry on oeis.org

1, 0, 3, -17, 169, -2079, 31261, -554483, 11336753, -262517615, 6791005621, -194103134499, 6074821125385, -206616861429575, 7588549099814957, -299320105069298459, 12619329503201165281, -566312032570838608863, 26952678355224681891685

Offset: 0

Joe Keane (jgk(AT)jgk.org), May 03 2002

sign

Inverse binomial transform of A000312. - Tilman Neumann, Dec 13 2008

The |a(n)| is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the digraph representation of f has no isolated vertices. - Geoffrey Critzer, Nov 13 2011

sci.math article 3CBC2B66.224E(AT)olympus.mons

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A086331.

Cf. A350212.

t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[Exp[-x]/(1 - t), {x, 0, 20}], x] (* Geoffrey Critzer, Nov 13 2011 *)
Range[0, 18]! CoefficientList[ Series[ Exp[x]/(1 + LambertW[x]), {x, 0, 18}], x] (* Robert G. Wilson v, Nov 28 2012 *)

my(x='x+O('x^20)); Vec(serlaplace(exp(x)/(1+lambertw(x)))) \\ G. C. Greubel, Jun 11 2017

a(n) = n! * Sum_{k=0..n} (-1)^k*k^k/(k!*(n - k)!).
E.g.f. for absolute value of {a(n)}: exp(C(x)-x) where C(x) is the e.g.f for A001865. - Geoffrey Critzer, Nov 13 2011, corrected by Vaclav Kotesovec, Nov 27 2012
abs(a(n)) ~ (exp(1)*n-1/2)/exp(1+exp(-1)) * n^(n-1). - Vaclav Kotesovec, Nov 27 2012
a(n) = (-1)^n * A350212(n,0). - Alois P. Heinz, Dec 19 2021

A277473 E.g.f.: -exp(x)*LambertW(-x).

Original entry on oeis.org

0, 1, 4, 18, 116, 1060, 12702, 187810, 3296120, 66897288, 1540762010, 39693752494, 1130866726596, 35300006582620, 1198036854980630, 43921652697277170, 1729775120233353968, 72831210167041246480, 3264674481128340280242, 155220967397580333229270

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000169, A069856, A086331, A277474.

Partial sums of A038051.

CoefficientList[Series[-Exp[x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[Binomial[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]

x='x+O('x^50); concat([0], Vec(serlaplace(-exp(x)*lambertw(-x)))) \\ G. C. Greubel, Jun 11 2017

a(n) = Sum_{k=1..n} binomial(n,k) * k^(k-1).

a(n) ~ exp(exp(-1)) * n^(n-1).

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

Original entry on oeis.org

1, 7, 105, 2386, 71890, 2695652, 120907185, 6312179764, 375971507406, 25160695768715, 1869031937691061, 152603843369288819, 13584174777196666630, 1309317592648179024666, 135850890740575408906465

Offset: 1

Amarnath Murthy, Jul 04 2004

nonn

The product of the terms of the n-th row is given by A034841.

Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016

			From _Seiichi Manyama_, Aug 18 2018: (Start)
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (1*2 + 3*4) = 7.
a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..338

Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.

List(List([1..20],n->List([1..n],k->Binomial(k*n,n))),Sum); # Muniru A Asiru, Aug 12 2018

A096130 := proc(n,k) binomial(k*n,n) ; end: A096131 := proc(n) local k; add( A096130(n,k),k=1..n) ; end: for n from 1 to 18 do printf("%d, ",A096131(n)) ; od ; # R. J. Mathar, Apr 30 2007
seq(add((binomial(n*k,n)), k=0..n), n=1..15); # Zerinvary Lajos, Sep 16 2007

Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 06 2013 *)

a(n) = sum(k=1, n, binomial(k*n, n)); \\ Michel Marcus, Aug 20 2018

a(n) = Sum_{k=1..n} binomial(k*n, n). - Reinhard Zumkeller, Jan 09 2005
a(n) = (1/n!) * Sum_{j=1..n} Product_{i=n*(j-1)+1..n*j} i. - Reinhard Zumkeller, Jan 09 2005 [corrected by Seiichi Manyama, Aug 17 2018]
a(n) ~ exp(1)/(exp(1)-1) * binomial(n^2,n). - Vaclav Kotesovec, Jun 06 2013

More terms from R. J. Mathar, Apr 30 2007

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

Original entry on oeis.org

1, 2, 9, 103, 2073, 58481, 2101813, 91492906, 4671050401, 273437232283, 18046800575211, 1325445408799007, 107200425419863009, 9466283137384124247, 906151826270369213655, 93459630239922214535911, 10331984296666203358431361, 1218745075041575200343722415

Offset: 0

Vaclav Kotesovec, Jun 06 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A072034, A086331, A096131, A167008, A167010, A349471.

[(&+[Binomial(n*j,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Aug 31 2022

Table[Sum[Binomial[k*n, k], {k, 0, n}], {n, 0, 20}]

A226391(n):=sum(binomial(k*n,k), k,0,n); makelist(A226391(n),n,0,30); /* Martin Ettl, Jun 06 2013 */

@CachedFunction
def A226391(n): return sum(binomial(n*j, j) for j in (0..n))
[A226391(n) for n in (0..30)] # G. C. Greubel, Aug 31 2022

a(n) ~ binomial(n^2, n).

A277452 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * k!.

Original entry on oeis.org

1, 2, 13, 226, 7889, 458026, 39684637, 4788052298, 766526598721, 157108817646514, 40104442275129101, 12472587843118746322, 4641978487740740993233, 2036813028164774540010266, 1040451608604560812273060189, 612098707457003526384666111226

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..232

Main diagonal of A320031.

Cf. A086331, A277373, A277423, A277453.

a := n -> simplify(hypergeom([1, -n], [], -n)):
seq(a(n), n=0..15); # Peter Luschny, Oct 03 2018
# second Maple program:
b:= proc(n, k) option remember;
      1 + `if`(n>0, k*n*b(n-1, k), 0)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 09 2020

Flatten[{1, Table[Sum[Binomial[n, k]*n^k*k!, {k, 0, n}], {n, 1, 20}]}]

a(n) = sum(k=0, n, binomial(n,k) * n^k * k!); \\ Michel Marcus, Sep 18 2018

a(n) = exp(1/n) * n^n * Gamma(n+1, 1/n).
a(n) ~ n^n * n!.
a(n) = n! * [x^n] exp(x)/(1 - n*x). - Ilya Gutkovskiy, Sep 18 2018
a(n) = floor(n^n*n!*exp(1/n)), n > 0. - Peter McNair, Dec 20 2021

A277458 Expansion of e.g.f. -1/(1-LambertW(-x)).

Original entry on oeis.org

-1, 1, 0, 3, 16, 165, 2016, 30415, 539904, 11049129, 256038400, 6627314331, 189517916160, 5933803272397, 201893195083776, 7417376809230375, 292648536838045696, 12341039738944113105, 553942486234823786496, 26369048375194607316019, 1326864458454400696320000

Offset: 0

Vaclav Kotesovec, Oct 16 2016

G. C. Greubel, Table of n, a(n) for n = 0..385
Jason Saied, Jeffrey Marshall, Namit Anand, and Eleanor G. Rieffel, General protocols for the efficient distillation of indistinguishable photons, arxiv:2404.14217 [quant-ph], Apr 22 2024. See p. 14.

Cf. A000312, A086331, A133297, A134095, A277510, A308506.

seq(n!*add((-1)^(k+1)*k*n^(n-k-1)/(n-k)!, k = 1..n), n = 1..20); # Peter Bala, Jul 23 2021

CoefficientList[Series[-1/(1-LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!

my(x='x+O('x^50)); Vec(serlaplace(-1/(1 - lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ n^(n-1) / 4.
a(n) = n!*Sum_{k = 1..n} (-1)^(k+1)*k*n^(n-k-1)/(n-k)! for n >= 1. Cf. A133297. - Peter Bala, Jul 23 2021
a(n) = (-1)^(n+1)*U(1-n, -n, -n) where U is the Kummer U function. - Peter Luschny, Jan 23 2025

A323280 a(n) = Sum_{k=0..n} binomial(n,k) * k^(2*k).

Original entry on oeis.org

1, 2, 19, 781, 68553, 10100761, 2236373953, 693667946945, 286962262702657, 152652510206521921, 101513694573289791441, 82511051259976074269425, 80480313356721971865934369, 92773167329045961244649105633, 124768226258051318899374299271601

Offset: 0

Seiichi Manyama, Jan 12 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. A062206, A064570, A086331, A242446, A277454, A277456, A308490, A316747.

Table[1 + Sum[Binomial[n, k]*k^(2*k), {k, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 31 2019 *)

a(n) = sum(k=0, n, binomial(n, k)*k^(2*k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k^2*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022

a(n) ~ n^(2*n). - Vaclav Kotesovec, May 31 2019
From Seiichi Manyama, Jul 04 2022: (Start)
G.f.: Sum_{k>=0} (k^2 * x)^k/(1 - x)^(k+1).
E.g.f.: exp(x) * Sum_{k>=0} (k^2 * x)^k/k!. (End)

cp's OEIS Frontend

A294353 Product of first n terms of the binomial transform of n^n (A086331).

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

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A088957 Hyperbinomial transform of the sequence of 1's.

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A069856 E.g.f.: exp(x)/(1+LambertW(x)).

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A277473 E.g.f.: -exp(x)*LambertW(-x).

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A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

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A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

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