A006675 -id:A006675 - cp's OEIS frontend

A001705 Generalized Stirling numbers: a(n) = n! * Sum_{k=0..n-1} (k+1)/(n-k).

0, 1, 5, 26, 154, 1044, 8028, 69264, 663696, 6999840, 80627040, 1007441280, 13575738240, 196287356160, 3031488633600, 49811492505600, 867718162483200, 15974614352793600, 309920046408806400, 6320046028584960000, 135153868608460800000, 3024476051557847040000

Offset: 0

N. J. A. Sloane

a(n) is also the sum of the positions of the right-to-left minima in all permutations of [n]. Example: a(3)=26 because the positions of the right-to-left minima in the permutations 123,132,213,231,312 and 321 are 123, 13, 23, 3, 23 and 3, respectively and 1 + 2 + 3 + 1 + 3 + 2 + 3 + 3 + 2 + 3 + 3 = 26. - Emeric Deutsch, Sep 22 2008
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=2) ~ exp(-x)/x^2*(1 - 5/x + 26/x^2 - 154/x^3 + 1044/x^4 - 8028/x^5 + 69264/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
a(n) is the total number of cycles (excluding fixed points) in all permutations of [n+1]. - Olivier Gérard, Oct 23 2012; Dec 31 2012
A length n sequence is formed by randomly selecting (one-by-one) n real numbers in (0,1).  a(n)/(n+1)! is the expected value of the sum of the new maximums in such a sequence.  For example for n=3: If we select (in this order): 0.591996, 0.646474, 0.163659  we would add 0.591996 + 0.646474 which would be a bit above the average of a(3)/4! = 26/24. - Geoffrey Critzer, Oct 17 2013

			(1-x)^-2 * (-log(1-x)) = x + 5/2*x^2 + 13/3*x^3 + 77/12*x^4 + ...
Examples: a(6) = 6!*(1/6 + 2/5 + 3/4 + 4/3 + 5/2 + 6/1) = 8028; a(20) = 20!*(1/20 + 2/19 + 3/18 + 4/17 + 5/16 + ... + 16/5 + 17/4 + 18/3 + 19/2 + 20/1) = 135153868608460800000. - _Alexander Adamchuk_, Oct 09 2004
From _Olivier Gérard_, Dec 31 2012: (Start)
The cycle decomposition of all permutations of 4 elements gives the following list: {{{1},{2},{3},{4}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4,3}}, {{1},{2,3,4}}, {{1},{2,4},{3}}, {{1,2},{3},{4}}, {{1,2},{3,4}}, {{1,3,2},{4}},{{1,4,3,2}}, {{1,3,4,2}}, {{1,4,2},{3}}, {{1,2,3},{4}}, {{1,2,4,3}},{{1,3},{2},{4}}, {{1,4,3},{2}}, {{1,3},{2,4}}, {{1,4,2,3}}, {{1,2,3,4}}, {{1,2,4},{3}}, {{1,3,4},{2}}, {{1,4},{2},{3}}, {{1,3,2,4}}, {{1,4},{2,3}}}.
Deleting the fixed points gives the following 26 items: {{3,4}, {2,3}, {2,4,3}, {2,3,4}, {2,4}, {1,2}, {1,2}, {3,4}, {1,3,2}, {1,4,3,2}, {1,3,4,2}, {1,4,2}, {1,2,3}, {1,2,4,3}, {1,3}, {1,4,3}, {1,3}, {2,4}, {1,4,2,3}, {1,2,3,4}, {1,2,4}, {1,3,4}, {1,4}, {1,3,2,4}, {1,4}, {2,3}}. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..445 (terms 0 through 100 were provided by T. D. Noe)
J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, preprint, 2016.
J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, 340(10) (2017), 2550-2558.
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021.
Gang Chen, Henrik Johansson, Fei Teng, and Tianheng Wang, Next-to-MHV Yang-Mills kinematic algebra, arXiv:2104.12726 [hep-th], 2021. See p. 46.
Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See p. 24.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 406.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49. [Annotated scanned copy]
Motohico Mulase, In Search of a Hidden Curve, arXiv:2501.00716 [math.QA], 2025. See p. 27.
J. Riordan, Letter of 04/11/74.
J. Riordan, Letter, Jul 06 1978.

Cf. A000254 (total number of cycles in permutations, including fixed points).
Cf. A002104 (number of different cycles in permutations, without fixed points).
Cf. A006231 (number of different cycles in permutations, including fixed points).
Cf. A001008, A002805, A006675, A066667, A132159, A143947.
Related to n!*the k-th successive summation of the harmonic numbers:
  (k=0) A000254, (k=1) A001705, (k=2) A001711, (k=3) A001716,
  (k=4) A001721, (k=5) A051524, (k=6) A051545, (k=7) A051560,
  (k=8) A051562, (k=9) A051564.

a := n-> add((n+1)!/k, k=2..n+1): seq(a(n), n=0..21); # Zerinvary Lajos, Jan 22 2008; edited Johannes W. Meijer, Nov 28 2012
a := n -> ((n+1)!*(h(n+1)-1)): h := n-> harmonic(n): seq(a(n), n=0..21); # Gary Detlefs, Dec 18 2009; corrected by Johannes W. Meijer, Nov 28 2012

Table[n!*Sum[Sum[1/k,{k,1,m}], {m,1,n}], {n,0,20}] (* Alexander Adamchuk, Apr 14 2006 *)
a[n_] := (n + 1)! (EulerGamma - 1 + PolyGamma[n + 2]);
Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 19 2022 *)

a(n):=n!*sum(((-1)^(k+1)*binomial(n+1,k+1))/k,k,1,n); /* Vladimir Kruchinin, Oct 10 2016 */

for(n=0,25, print1(n!*sum(k=0,n-1,(k+1)/(n-k)), ", ")) \\ G. C. Greubel, Jan 20 2017

from math import factorial
def A001705(n):
    f = factorial(n)
    return sum(f*(k+1)//(n-k) for k in range(n)) # Chai Wah Wu, Jun 23 2022

Partial sum of first n harmonic numbers multiplied by n!.
a(n) = n!*Sum_{m=1..n} Sum_{k=1..m} 1/k = n!*Sum_{m=1..n} H(m), where H(m) = Sum_{k=1..m} 1/k = A001008(m)/A002805(m) is m-th Harmonic number.
E.g.f.: - log (1 - x) / (1 - x)^2.
a(n) = (n+1)! * H(n) - n*n!, H(n) = Sum_{k=1..n} (1/k).
a(n) = A112486(n, 1).
a(n) = a(n-1)*(n+1) + n! = A000254(n+1) - A000142(n+1) = A067176(n+1, 1). - Henry Bottomley, Jan 09 2002
a(n) = Sum_{k=0..n-1} ((-1)^(n-1+k) * (k+1) * 2^k * Stirling1(n, k+1)). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at 0. - Ralf Stephan, Apr 16 2004
a(n) = Sum_{k=1..n} (k*StirlingCycle(n+1,k+1)). - David Callan, Sep 25 2006
a(n) = Sum_{k=n..n*(n+1)/2} k*A143947(n,k). - Emeric Deutsch, Sep 22 2008
For n >= 1, a(n) = Sum_{j=0..n-1} ((-1)^(n-j-1) * 2^j * (j+1) * Stirling1(n,j+1)). - Milan Janjic, Dec 14 2008
a(n) = (2*n+1)*a(n-1) - n^2*a(n-2). - Gary Detlefs, Nov 27 2009
a(n) = (n+1)!*(H(n+1) - 1) where H(n) is the n-th harmonic number. - Gary Detlefs, Dec 18 2009
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n+1,k+1)/k. - Vladimir Kruchinin, Oct 10 2016
a(n) = (n+1)!*Sum_{k = 1..n} (-1)^(k+1)*binomial(n+1,k+1)*k/(k+1). - Peter Bala, Feb 15 2022
a(n) = Gamma(n + 2) * (Digamma(n + 2) + EulerGamma - 1). - Peter Luschny, Feb 19 2022
From Mélika Tebni, Jun 22 2022: (Start)
a(n) = -Sum_{k=0..n} k!*A066667(n, k+1).
a(n) = Sum_{k=0..n} k!*A132159(n, k+1). (End)
a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], 1)/2. - Peter Luschny, Jun 22 2022

More terms from Sascha Kurz, Mar 22 2002

A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.

Original entry on oeis.org

1, 5, 13, 77, 87, 223, 481, 4609, 4861, 55991, 58301, 785633, 811373, 835397, 1715839, 29889983, 30570663, 197698279, 201578155, 41054655, 13920029, 325333835, 990874363, 25128807667, 25472027467, 232222818803, 235091155703, 6897956948587, 6975593267347

Offset: 1

Glen Burch (gburch(AT)erols.com)

Numerator of a second-order harmonic number H(n, (2)) = Sum_{k=1..n} HarmonicNumber(k). - Alexander Adamchuk, Apr 12 2006
p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 06 2006
Denominator is A027611(n+1). p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 26 2006
a(n) = A213998(n,n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A001705, A002805, A006675, A022819, A027611, A093418.

import Data.Ratio ((%), numerator)
a027612 n = numerator $ sum $ zipWith (%) [1 .. n] [n, n-1 .. 1]
-- Reinhard Zumkeller, Jul 03 2012

[Numerator((&+[j/(n-j+1): j in [1..n]])): n in [1..30]]; // G. C. Greubel, Aug 23 2022

a := n -> numer(add((n+1-j)/j, j=1..n));
seq(a(n), n = 1..29); # Peter Luschny, May 12 2023

Numerator[Table[Sum[Sum[1/i,{i,1,k}],{k,1,n}],{n,1,30}]] (* Alexander Adamchuk, Apr 12 2006 *)
Numerator[Table[Sum[k/(n-k+1),{k,1,n}],{n,1,50}]] (* Alexander Adamchuk, Jul 26 2006 *)

a(n) = numerator(sum(k=1, n, k/(n-k+1))); \\ Michel Marcus, Jul 14 2018

[numerator(n*(harmonic_number(n+1) - 1)) for n in (1..30)] # G. C. Greubel, Aug 23 2022

From Vladeta Jovovic, Sep 02 2002: (Start)
a(n) = numerators of coefficients in expansion of -log(1-x)/(1-x)^2.
a(n) = numerators of (n+1)*(harmonic(n+1) - 1).
a(n) = numerators of (n+1)*(Psi(n+2) + Euler-gamma - 1). (End)
a(n) = numerator( Sum_{k=1..n} Sum_{i=1..k} 1/i ). - Alexander Adamchuk, Apr 12 2006
a(n) = numerator( Sum_{k=1..n} k/(n-k+1) ). - Alexander Adamchuk, Jul 26 2006
a(n) = numerator of integral_{x=1..n+1} floor((n+1)/x). - Jean-François Alcover, Jun 18 2013

A027611 Denominator of n * n-th harmonic number.

Original entry on oeis.org

1, 1, 2, 3, 12, 10, 20, 35, 280, 252, 2520, 2310, 27720, 25740, 24024, 45045, 720720, 680680, 4084080, 3879876, 739024, 235144, 5173168, 14872858, 356948592, 343219800, 2974571600, 2868336900, 80313433200, 77636318760

Offset: 1

Glen Burch (gburch(AT)erols.com)

This is very similar to A128438, which is a different sequence. They differ at n=6 (and nowhere else?). - N. J. A. Sloane, Nov 21 2008
Denominator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.
Denominator of Sum_{k=1..n} frac(n/k) where frac(x/y) denotes the fractional part of x/y. - Benoit Cloitre, Oct 03 2002
Denominator of Sum_{d=2..n-1, n mod d > 0} n/d. Numerator = A079076. - Reinhard Zumkeller, Dec 21 2002
a(n) is odd iff n is a power of 2. - Benoit Cloitre, Oct 03 2002
Indices where a(n) differs from A128438 are terms of A074791. - Gary Detlefs, Sep 03 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Complete Set

Harmonic numbers = A001008/A002805.
Cf. A001705, A006675, A027612, A049820, A024816.
Cf. A128438, A074791, A079076.

import Data.Ratio ((%), denominator)
a027611 n = denominator $ sum $ map (n %) [1..n]
-- Reinhard Zumkeller, Jul 03 2012

[Denominator(n*HarmonicNumber(n)): n in [1..40]]; // Vincenzo Librandi, Feb 19 2014

a := n -> denom(add((n-j)/j, j=1..n));
seq(a(n), n = 1..30); # Peter Luschny, May 12 2023

a[n_]:=Denominator[n*HarmonicNumber[n]]; Array[a,100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)

a(n) = denominator(n*sum(k=1, n, 1/k)); \\ Michel Marcus, Feb 15 2015

from sympy import harmonic
def A027611(n): return (n*harmonic(n)).q # Chai Wah Wu, Sep 26 2021

[denominator(n*harmonic_number(n)) for n in (1..40)] # G. C. Greubel, Aug 24 2022

From Vladeta Jovovic, Sep 02 2002: (Start)
a(n) = denominators of coefficients in expansion of -log(1-x)/(1-x)^2.
a(n) = denominators of (n+1)*(harmonic(n+1) - 1).
a(n) = denominators of (n+1)*(Psi(n+2) + Euler-gamma - 1). (End)
a(n) = numerator(h(n)/h(n-1)) - denominator(h(n)/h(n-1)), n > 1, where h(n) is the n-th harmonic number. - Gary Detlefs, Sep 03 2011
a(n) = A213999(n, n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012
a(n) = denominators of coefficients of e.g.f. -1 + exp(x)*(1 + Sum_{j >= 0} (-x)^(j+1)/(j * j!)). - G. C. Greubel, Aug 24 2022

Entry revised by N. J. A. Sloane following a suggestion of Eric W. Weisstein, Jul 02 2004

A022819 a(n) = floor(1/(n-1) + 2/(n-2) + 3/(n-3) + ... + (n-1)/1).

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 8, 11, 13, 16, 19, 22, 25, 28, 31, 34, 38, 41, 44, 48, 51, 55, 59, 62, 66, 70, 74, 78, 81, 85, 89, 93, 97, 101, 106, 110, 114, 118, 122, 126, 131, 135, 139, 144, 148, 152, 157, 161, 166, 170, 174, 179, 183, 188, 193, 197, 202, 206, 211, 216

Offset: 0

Clark Kimberling

nonn

a(n) = A214075(n,n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012

			a(2) = floor(1/1) = 1;
a(3) = floor(1/2 + 2/1) = floor(5/2) = 2;
a(4) = floor(1/3 + 2/2 + 3/1) = floor(26/6) = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A027612.

import Data.Ratio ((%))
a022819 n = floor $ sum $ zipWith (%) [1 .. n-1] [n-1, n-2 .. 1]
-- Reinhard Zumkeller, Jul 03 2012

s=0; Table[s+=HarmonicNumber[j]//N; Floor[s],{j,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2010 *)
Join[{0},Floor[Accumulate[HarmonicNumber[Range[0,60]]]]] (* Harvey P. Dale, Sep 16 2019 *)

a(n) = floor(sum_{i=2..n} n/i) = floor(A000027(n)*(A001008(n)/A002805(n)-1)) = floor(A006675(n)/A000142(n)) = floor(A001705(n-1)/A000142(n-1)). - Henry Bottomley, May 05 2001

A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

Original entry on oeis.org

1, 2, 2, 6, 15, 9, 24, 104, 144, 64, 120, 770, 1775, 1750, 625, 720, 6264, 20880, 33480, 25920, 7776, 5040, 56196, 250096, 571095, 708295, 453789, 117649, 40320, 554112, 3127040, 9433088, 16486400, 16744448, 9175040, 2097152, 362880, 5973264, 41229324, 156498804

Offset: 1

Peter Bala, Nov 16 2015

Related to A220883 and A251592.

			Triangle begins
...1
...2      2
...6     15       9
..24    104     144      64
.120    770    1775    1750     625
.720   6264   20880   33480   25920    7776
5040  56196  250096  571095  708295  453789  117649
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998

P. Bala, Fractional iteration of a series inversion operator

A000142 (column 0), A000169 (main diagonal), A006675 (column 1). Cf. A001700, A025174, A056856, A163456, A220883, A224274, A251592.

seq(seq(coeff(mul(n*t + k, k = 2 .. n), t, i), i = 0..n-1), n = 1..10);

E.g.f. (with constant term 1 included): A(x,t) = [ 1/x*Revert( x*(1 - x)^t ) ]^(1/t) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + n,n)*x^n = 1 + x + (2 + 2*t)*x^2/2! + (2 + 3*t)*(3 + 3*t)*x^3/3! + (2 + 4*t)*(3 + 4*t)*(4 + 4*t)*x^4/4! + ..., where Revert denotes the series reversion operator with respect to x.
In the notation of the Bala link, A(x,t) = I^t(1/(1 - x)) where I^t is a fractional inversion operator.
A(x,t) = B_(1+t)(x), where B_t(x) is the e.g.f. for A251592 and is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.
A(x,t)^m = Sum_{n >= 0} m/(n*t + m)*binomial(n*t + n + m - 1,n)*x^n = 1 + m*x + m*(2*t + m + 1)*x^2/2! + m*(3*t + m + 1)*(3*t + m + 2)*x^3/3! + m*(4*t + m + 1)*(4*t + m + 2)*(4*t + m + 3)*x^4/4! + ....
A(x,t)^t = 1 + t*x + t(1 + 3*t)*x^2/2! + t*(1 + 4*t)*(2 + 4*t)*x^3/3! + t*(1 + 5*t)*(2 + 5*t)*(3 + 5*t)*x^4/4! + ... is the e.g.f for A220883 with an extra constant term 1 and an extra factor of t included.
t*log( A(x,t) ) = t*x + t*(1 + 2*t)*x^2/2! + t*(1 + 3*t)*(2 + 3*t)*x^3/3! + t*(1 + 4*t)*(2 + 4*t)*(3 + 4*t)*x^4/4! + ... is the e.g.f for A056856.
For n = 1,2,3,..., the sequence [x^n] A(x,t)^n = [1, (2*t + 3), (3*t + 4)*(3*t + 5)/2!, (4*t + 5)*(4*t + 6)*(4*t + 7)/3!, ...]. This sequence has the following specializations:
t = 0: [1, 3, 10, 35, 126, ...] = A001700 (with different offset).
t = 1: [1, 5, 28, 165, 1001, ...] = A025174.
t = 2: [1, 7, 55, 455, 3876, ...] = A224274.
t = 3: [1, 9, 91, 969, 10626, ...] = A163456.

A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 2, 2, 3, 2, 4, 3, 1, 2, 5, 3, 3, 2, 2, 1, 3, 3, 3, 1, 1, 2, 2, 2, 5, 2, 2, 2, 5, 1, 3, 4, 4, 3, 3, 3, 5, 4, 3, 3, 3, 2, 2, 6, 2, 3, 4, 2, 4, 2, 3, 3, 2, 4, 4, 4, 3, 3, 3, 3, 4, 2, 3, 4, 2, 2, 5, 3, 2, 2, 4, 4, 2, 2, 1, 6, 4, 2, 5, 3, 5, 1, 2, 2, 3, 4, 2, 3, 3, 3, 5

Offset: 1

Jonathan Vos Post, Nov 09 2006

We must include multiplicity in the definition due to terms such as a(16) = 29889983 = 19 * 31^2 * 1637. The primes are those n for which a(n) = Omega(A027612(n))= 1, namely a(2) = 5, a(3) = 13, a(6) = 223, a(9) = 4861, a(18) = 197698279, a(25) = 25472027467. The semiprimes are those for which a(n) = 2, such as when n = 4, 5, 7, 8, 11, 12, 13, 15, 19, 23, 24. The 3-almost primes are those for which a(n) = 3, as with the "3-brilliant" a(10) = 55991 = 13 * 59 * 73, a(14), a(17), a(21), a(22), a(26).

			a(20) = 5 because A027612(20) = 41054655 = 3 * 5 * 23 * 127 * 937 has 5 prime factors.

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, pp. 143 and 258-259.

G. C. Greubel, Table of n, a(n) for n = 1..150
Eric Weisstein's World of Mathematics, Harmonic Number, MathWorld, see discussion of Conway and Guy (1996) definition of the second-order harmonic number.

Cf. A001222 Number of prime divisors of n (counted with multiplicity), A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n, A027611, A001008, A002805, A001705, A006675, A093418.

PrimeOmega[Numerator[Table[Sum[k/(n - k + 1), {k, 1, n}], {n, 1, 50}]]] (* G. C. Greubel, Jan 22 2017 *)

a(n) = A001222(A027612(n)) = Omega(Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n).

A300910 Expansion of e.g.f. 1/(1 - x)^(x/(1 - x)^2).

Original entry on oeis.org

1, 0, 2, 15, 116, 1070, 11754, 149436, 2145296, 34193736, 598061160, 11377384920, 233732130312, 5153974126704, 121354505626704, 3037419444974040, 80497938647953920, 2251124265581428800, 66225476356207660224, 2044005966844402035456, 66025689709572751040640, 2227221130525199246067840, 78301158190416233445985920

Offset: 0

Ilya Gutkovskiy, Mar 15 2018

nonn

Exponential transform of A006675.

			1/(1 - x)^(x/(1 - x)^2) = 1 + 2*x^2/2! + 15*x^3/3! + 116*x^4/4! + 1070*x^5/5! + 11754*x^6/6! + 149436*x^7/7! + ...

N. J. A. Sloane, Transforms

Cf. A000254, A001705, A006675, A027611, A027612, A087761, A300491.

a:=series(1/(1-x)^(x/(1-x)^2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[1/(1 - x)^(x/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k k! (HarmonicNumber[k] - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

E.g.f.: A(x) = exp(B(x)*C(x)), where B(x) is the g.f. of the sequence {0, 1, 2, 3, 4, 5, ...} and C(x) is the g.f. of the sequence {0, 1, 1/2, 1/3, 1/4, 1/5, ...}.

a(0) = 1; a(n) = Sum_{k=1..n} k*k!*(H(k)-1)*binomial(n-1,k-1)*a(n-k), where H(k) is the k-th harmonic number.

cp's OEIS Frontend

A001705 Generalized Stirling numbers: a(n) = n! * Sum_{k=0..n-1} (k+1)/(n-k).

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A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.

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A027611 Denominator of n * n-th harmonic number.

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A022819 a(n) = floor(1/(n-1) + 2/(n-2) + 3/(n-3) + ... + (n-1)/1).

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A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

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A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

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