A095996 -id:A095996 - cp's OEIS frontend

A096002 Let {b(n)} = A095996; then a(n) = number of terms {b(1),b(2),...,b(n-1)} which are coprime to n.

0, 1, 2, 2, 4, 2, 6, 4, 4, 3, 10, 3, 12, 4, 3, 8, 16, 4, 18, 4, 4, 6, 22, 5, 8, 7, 10, 5, 28, 2, 30, 16, 5, 9, 5, 7, 36, 10, 6, 6, 40, 3, 42, 7, 5, 12, 46, 9, 12, 7, 7, 8, 52, 10, 6, 7, 8, 15, 58, 3, 60, 16, 6, 32, 6, 3, 66, 10, 9, 3, 70, 13, 72, 19, 7, 11, 7, 4, 78, 10

Offset: 1

Robert G. Wilson v, Jul 19 2004, based on a suggestion from Leroy Quet, Jun 18 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A095996.

f[n_] := f[n] = Select[Divisors[n! ], GCD[ #, n] == 1 &][[ -1]]; Table[ Length[ Select[ Range[n - 1], GCD[ f[ # ], n] == 1 &]], {n, 2, 33}]

seq(n)={my(v=vector(n, n, denominator(n^n/n!))); vector(n, n, sum(k=1, n-1, gcd(v[k],n)==1));} \\ Andrew Howroyd, Nov 05 2019

If p is prime, then a(p) = p-1.

a(33) - a(41) from Robert G. Wilson v, May 05 2013

a(1) corrected and terms a(42) and beyond from Andrew Howroyd, Nov 05 2019

A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 9, -6, 1, 0, -64, 48, -12, 1, 0, 625, -500, 150, -20, 1, 0, -7776, 6480, -2160, 360, -30, 1, 0, 117649, -100842, 36015, -6860, 735, -42, 1, 0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1, 0, 43046721, -38263752, 14880348, -3306744, 459270, -40824, 2268, -72, 1

Offset: 0

Roger L. Bagula, Apr 18 2008

Row sums give A177885.
The Abel polynomials are associated with the Abel operator t*exp(y*t)*p(x) = t*p(x+y).
From Peter Luschny, Jan 14 2009: (Start)
Abs(T(n,k)) is the number of rooted labeled trees on n+1 vertices with a root degree k (Clarke's formula).
The row sums in the unsigned case, Sum_{k=0..n} abs(T(n,k)), count the trees on n+1 labeled nodes, A000272(n+1). (End)
Exponential Riordan array [1, W(x)], W(x) the Lambert W-function. - Paul Barry, Nov 19 2010
The inverse array is the exponential Riordan array [1, x*exp(x)], which is A059297. - Peter Bala, Apr 08 2013
The inverse Bell transform of [1,2,3,...]. See A264428 for the Bell transform and A264429 for the inverse Bell transform. - Peter Luschny, Dec 20 2015
Also the Bell transform of (-1)^n*(n+1)^n. - Peter Luschny, Jan 18 2016

			Triangle begins:
  1;
  0,        1;
  0,       -2,       1;
  0,        9,      -6,       1;
  0,      -64,      48,     -12,      1;
  0,      625,    -500,     150,    -20,      1;
  0,    -7776,    6480,   -2160,    360,    -30,    1;
  0,   117649, -100842,   36015,  -6860,    735,  -42,   1;
  0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29

Seiichi Manyama, Rows n = 0..139, flattened
W. Y. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
L. E. Clarke, On Cayley's formula for counting trees, J. London Math. Soc. 33 (1958), 471-475.
Péter L. Erdős and L. A. Székely, Applications of Antilexicographic Order. I., An Enumerative Theory of Trees, Adv. in Appl. Math. 10, (1989) 488-496.
Eric Weisstein's World of Mathematics, Abel Polynomial.
Wikipedia, Abel Polynomials.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Row sums A177885.
Cf. A000272, A061356, A059297 (inverse array), A264429.
Cf. A061354, A095996, A264234, A061355, A049444.

T := proc(n,k) if n = 0 and k = 0 then 1 else binomial(n-1,k-1)*(-n)^(n-k) fi end; seq(print(seq(T(n,k),k=0..n)),n=0..7); # Peter Luschny, Jan 14 2009
# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-n-1)^n, 9); # Peter Luschny, Jan 27 2016

a0 = 1 a[x, 0] = 1; a[x, 1] = x; a[x_, n_] := x*(x - a0*n)^(n - 1); Table[Expand[a[x, n]], {n, 0, 10}]; a1 = Table[CoefficientList[a[x, n], x], {n, 0, 10}]; Flatten[a1]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, (-n-1)^n], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

# uses[inverse_bell_transform from A264429]
def A137452_matrix(dim):
    nat = [n for n in (1..dim)]
    return inverse_bell_transform(dim, nat)
A137452_matrix(10) # Peter Luschny, Dec 20 2015

Row n gives the coefficients of the expansion of x*(x-n)^(n-1).
Abs(T(n,k)) = C(n-1,k-1)*n^(n-k). - Peter Luschny, Jan 14 2009
From Wolfdieter Lang, Nov 08 2022: (Start)
From the exponential Riordan (also Sheffer of Jabotinsky) type (1, LambertW) array (see comments).
E.g.f. of column sequence k, LambertW(x)^k/k!, for k >= 0.
E.g.f. of row polynomials P_n(y) = Sum_{k=0..n} T(n, k)*y^k: exp(y*LambertW(x)).
Recurrence for T: T(n, k) = 0 for n < k; T(n, 0) = 1 for n = 0 otherwise 0; T(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j,k-1)*(-1)^j*T(n-1, k-1+j). (Jabotinsky type convolution triangle, the e.g.f.s for the a- and z-sequences are exp(-x), and 0. See the link in A006232.)
Recurrence for column k of T: T(n, k) = 0 for n < k, T(k, k) = 1, for k >= 0 otherwise T(n, k) = (n!*k/(n-k))*Sum_{j=k..n-1} (1/j!)*beta(n-1-j)*T(j, k), where beta(n) = A264234(n+1)/A095996(n+1) = {-1, 2, -9/2, 32/3, -625/24, ...} with o.g.f. d/dx(log(LambertW(x)/x)). See the Boas-Buck or Rainville references given in A046521, and my Aug 10 2017 comment there.
Recurrence for the row polynomials P_0(x) = 1, and P_n(x) = x*substitute(z=d/dx, exp(-z)/(1+z)) P_(n-1)(x), for n >= 1, with coefficient z^k of exp(-z)/(1+z) given by (-1)^k*A061354(k)/A061355(k). See the Roman reference Corollary 3.7.2., p. 50. (End)
The column sequences for the unsigned triangle Abs(T(n, k)), for k >= 2, are also given by {n^(n-k)*(n-1)*s(k-2, n)/(k-1)!}A049444.%20-%20_Wolfdieter%20Lang">{n>=k} with the row polynomials s(n, x) = risingfactorial(x - (n+1), n) of A049444. - _Wolfdieter Lang, Nov 21 2022

Better name by Peter Bala, Apr 08 2013

Edited by Joerg Arndt, Apr 08 2013

A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

Original entry on oeis.org

2, 5, 26, 103, 2194, 1223, 472730, 556403, 21323986, 7281587, 125858034202, 180451625, 121437725363954, 595953719897, 26649932810926, 3211211914492699, 285050975993898158530, 549689343118061, 640611888918574971191834

Offset: 1

Robert G. Wilson v, Feb 13 2004

Also numerators of e_n(n) where e_n(x) is the exponential sum function exp_n(x) and where denominators are given by either A095996 (largest divisor of n! that is coprime to n) or A036503 (denominator of n^(n-2)/n!). - Gerald McGarvey, Nov 14 2005
a(n) is a multiple of A120266(n) or equals A120266(n), A120266(n) is numerator of Sum_{k=0..n} n^k/k!, the integral = (n-1)!/n^(n-1) * the Sum. - Gerald McGarvey, Apr 17 2008
The integral = (1/n^n)*A063170[n] (Schenker sums with n-th term, Integral_{x>0} exp(-x)*(n+x)^n dx). - Gerald McGarvey, Apr 17 2008
Expected value in the birthday paradox problem.  Let X be a random variable that assigns to each f:{1,2,...,n+1}->{1,2,...,n} the smallest k in {2,3,...,n+1} such that f(k)=f(j) for some j < k.  a(n)/A036505(offset=1) = E(X) the expected value of X.  For n=365 E(X) is (surprising low) approximately 24. - Geoffrey Critzer, May 18 2013
Also numerator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. N. J. A. Sloane, Jul 31 2013

G. C. Greubel, Table of n, a(n) for n = 1..250
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
Eric Weisstein, Exponential Sum Function

Denominators are in A036505.

Cf. A120266, A063170.

[Numerator((&+[Binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k): k in [0..n]])): n in [1..20]]; // G. C. Greubel, Feb 08 2019

f[n_]:= Integrate[E^(-x)*(1+x/n)^n, {x,0,Infinity}]; Table[Numerator[ f[n]], {n, 1, 20}]
Table[Numerator[1 + Sum[If[k==0,1,Binomial[n,k]*(k/n)^k*((n-k)/n)^(n-k)], {k,0,n-1}]], {n,1,20}] (* G. C. Greubel, Feb 08 2019 *)

vector(20, n, numerator(sum(k=0, n, binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k)))) \\ G. C. Greubel, Feb 08 2019

[numerator(sum(binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) for k in (0..n))) for n in (1..20)] # G. C. Greubel, Feb 08 2019

a(n) = A036505(n-1)*Sum_{k=0..n} (A128433(n)/A128434(n)). - Reinhard Zumkeller, Mar 03 2007

Definition corrected by Gerald McGarvey, Apr 17 2008

A036505 Numerator of (n+1)^n/n!.

Original entry on oeis.org

1, 2, 9, 32, 625, 324, 117649, 131072, 4782969, 1562500, 25937424601, 35831808, 23298085122481, 110730297608, 4805419921875, 562949953421312, 48661191875666868481, 91507169819844, 104127350297911241532841, 640000000000000000, 865405750887126927009

Offset: 0

N. J. A. Sloane

Also denominator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. - N. J. A. Sloane, Jul 31 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.

Cf. A036503, A063170.

Cf. A095996 (denominators).

List([0..20], n -> NumeratorRat((n+1)^n/Factorial(n))); # Muniru A Asiru, Feb 12 2018

[Numerator((n+1)^n/Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Sep 10 2013

a:=n -> numer((n+1)^n/factorial(n)):  A036505 := [seq(a(n), n=0..20)]; # Muniru A Asiru, Feb 12 2018

CoefficientList[Series[1/(1 + ProductLog[-x]), {x, 0, 21}], x] // Numerator // Rest (* Jean-François Alcover, Feb 04 2013, after Vladimir Kruchinin *)

my(x='x+O('x^30)); apply(x -> numerator(x), Vec(-1+1/(1+lambertw(-x)))) \\ G. C. Greubel and Michel Marcus, Feb 08 2019

[numerator((n+1)^n/factorial(n)) for n in (0..20)] # G. C. Greubel, Feb 08 2019

a(n) = A090878(n+1)/Sum_{k=0..n+1} (A128433(n+1)/A128434(n+1)). - Reinhard Zumkeller, Mar 03 2007
G.f.: -x*e^(-LambertW(-x))/((LambertW(-x)+1)*LambertW(-x)). - Vladimir Kruchinin, Feb 04 2013
A simpler g.f. is 1/(1 + LambertW(-x)). - Jean-François Alcover, Feb 04 2013

A036504 Numerator of n^(n-1)/n!.

Original entry on oeis.org

1, 1, 3, 8, 125, 54, 16807, 16384, 531441, 156250, 2357947691, 2985984, 1792160394037, 7909306972, 320361328125, 35184372088832, 2862423051509815793, 5083731656658, 5480386857784802185939, 32000000000000000, 41209797661291758429

Offset: 1

N. J. A. Sloane

(-1)^(n+1)* n-th numerator of coefficient in LambertW(x) power series, where LambertW(x) is the transcendental function satisfying LambertW(x)*exp( LambertW(x) )=x. - Benoit Cloitre, May 08 2002

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A036503 (denominators).

A095996 is a better version of the denominators of the LambertW function. - N. J. A. Sloane, Jan 08 2021

[Numerator(n^(n-1)/Factorial(n)): n in [1..30]]; // G. C. Greubel, Sep 09 2018

Numerator[Table[n^(n - 1)/n!, {n, 1, 30}]] (* G. C. Greubel, Sep 09 2018 *)
CoefficientList[-LambertW[-x]/x + O[x]^21, x] // Numerator (* Jean-François Alcover, Jan 15 2019 *)

for(n=1, 30, print1(numerator(n^(n-1)/n!), ", ")) \\ G. C. Greubel, Sep 09 2018

[numerator(n^(n-1)/factorial(n)) for n in (1..30)] # G. C. Greubel, Feb 08 2019

A227831 Numerators of coefficients in Taylor series for LambertW(x).

Original entry on oeis.org

0, 1, -1, 3, -8, 125, -54, 16807, -16384, 531441, -156250, 2357947691, -2985984, 1792160394037, -7909306972, 320361328125, -35184372088832, 2862423051509815793, -5083731656658, 5480386857784802185939, -32000000000000000, 41209797661291758429, -244636361793658185164

Offset: 0

N. J. A. Sloane, Aug 01 2013

The denominators are 1, 1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, ..., which is A095996 prefixed by 1.

			0, 1, -1, 3/2, -8/3, 125/24, -54/5, 16807/720, -16384/315, 531441/4480, -156250/567, 2357947691/3628800, -2985984/1925, ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 34.

Alois P. Heinz, Table of n, a(n) for n = 0..300
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W Function
Eric Weisstein's World of Mathematics, Lambert W-Function
Wikipedia, Lambert W function

Cf. A095996. See also A036504/A036503.

series(LambertW(x),x,30); # N. J. A. Sloane, Jan 08 2021

Numerator[CoefficientList[Series[LambertW[x], {x, 0, 22}], x]] (* Mats Granvik, Nov 24 2013 *)
Numerator[CoefficientList[InverseSeries[Series[x/Sum[((-x)^n)/Factorial[n], {n, 0, 22}], {x, 0, 22}]], x]] (* Mats Granvik, Nov 24 2013 *)

Numerators of series reversion of x/(Sum_{n=0..infinity} ((-x)^n)/n!). - Mats Granvik, Nov 24 2013

A279911 a(n) = Sum_{i=1..n} denominator(n^i/i).

Original entry on oeis.org

0, 1, 2, 4, 6, 11, 10, 22, 22, 31, 28, 56, 36, 79, 58, 72, 86, 137, 80, 172, 112, 145, 148, 254, 146, 261, 208, 274, 230, 407, 182, 466, 342, 375, 360, 448, 322, 667, 456, 528, 444, 821, 384, 904, 592, 635, 676, 1082, 574, 1051, 692, 924, 836, 1379, 732, 1154, 912, 1153

Offset: 0

Wesley Ivan Hurt, Dec 22 2016

Daniel Suteu, Table of n, a(n) for n = 0..10000

Cf. A279912.

[0] cat [&+[Denominator(n^i/i):i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Jul 29 2019

A279911:=n->add(denom(n^i/i), i=1..n): seq(A279911(n), n=0..100);

a(n) = sum(k=1, n, if(gcd(n,k) == 1, k, denominator(n^k/k))); \\ Daniel Suteu, Jul 28 2019

a(n) = sum(k=1, n, if(gcd(n,k) == 1, k, vecmax(select(d->gcd(d, n) == 1, divisors(k))))); \\ Daniel Suteu, Jul 28 2019

a(n) = my(f=factor(n)[,1]); sum(k=1, n, if(gcd(n, k) == 1, k, gcd(vector(#f, j, k / f[j]^valuation(k, f[j]))))); \\ Daniel Suteu, Jul 29 2019

From Daniel Suteu, Jul 28 2019: (Start)
a(prime(n)) = A072205(n).
a(p^k) = (p^(2*k+1) + p + 2) / (2*(p+1)), for prime powers p^k.
a(n) = Sum_{k=1..n} gcd(m, k), where m = A095996(n).
a(n) = Sum_{k=1..n} f(n,k), where f(n,k) is the largest divisor d of k for which gcd(d, n) = 1. (End)
a(n) = Sum_{1<=k<=n, gcd(n,k)=1} phi(k)*floor(n/k). - Ridouane Oudra, May 24 2023

A097905 Triangle where a(m,n) = largest divisor of m! coprime to n.

Original entry on oeis.org

1, 2, 1, 6, 3, 2, 24, 3, 8, 3, 120, 15, 40, 15, 24, 720, 45, 80, 45, 144, 5, 5040, 315, 560, 315, 1008, 35, 720, 40320, 315, 4480, 315, 8064, 35, 5760, 315, 362880, 2835, 4480, 2835, 72576, 35, 51840, 2835, 4480, 3628800, 14175, 44800, 14175, 145152, 175

Offset: 1

Leroy Quet, Sep 04 2004, Apr 10 2007

Right edge of triangle is sequence A095996.

			a(6,3) = 80 because 80 is largest divisor of 720 which is coprime to 3.

Cf. A095996.

seq(seq(denom(m^n/n!),m=1..n),n=1..11); # Vladeta Jovovic, May 03 2005

Flatten[Table[Table[Select[Divisors[m! ], GCD[ #,n]==1&][[ -1]], {n,1,m}],{m,1, 10}]] (* Stefan Steinerberger, Nov 09 2007 *)

More terms from Stefan Steinerberger, Nov 09 2007

cp's OEIS Frontend

A096002 Let {b(n)} = A095996; then a(n) = number of terms {b(1),b(2),...,b(n-1)} which are coprime to n.

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A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).

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A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

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A036505 Numerator of (n+1)^n/n!.

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A036504 Numerator of n^(n-1)/n!.

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A227831 Numerators of coefficients in Taylor series for LambertW(x).

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