cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.

Original entry on oeis.org

0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880, 19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200, 22376988058521600, 431565146817638400, 8752948036761600000, 186244810780170240000
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Number of permutations of n+1 elements with exactly two cycles.
Number of cycles in all permutations of [n]. Example: a(3) = 11 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) have 11 cycles altogether. - Emeric Deutsch, Aug 12 2004
Row sums of A094310: In the symmetric group S_n, each permutation factors into k independent cycles; a(n) = sum k over S_n. - Harley Flanders (harley(AT)umich.edu), Jun 28 2004
The sum of the top levels of the last column over all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, the levels of their last columns being 2 and 1, respectively. - Emeric Deutsch, Aug 12 2006
a(n) is divisible by n for all composite n >= 6. a(2*n) is divisible by 2*n + 1. - Leroy Quet, May 20 2007
For n >= 2 the determinant of the n-1 X n-1 matrix M(i,j) = i + 2 for i = j and 1 otherwise (i,j = 1..n-1). E.g., for n = 3 the determinant of [(3, 1), (1, 4)]. See 53rd Putnam Examination, 1992, Problem B5. - Franz Vrabec, Jan 13 2008, Mar 26 2008
The numerator of the fraction when we sum (without simplification) the terms in the harmonic sequence. (1 + 1/2 = 2/2 + 1/2 = 3/2; 3/2 + 1/3 = 9/6 + 2/6 = 11/6; 11/6 + 1/4 = 44/24 + 6/24 = 50/24;...). The denominator of this fraction is n!*A000142. - Eric Desbiaux, Jan 07 2009
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=1) ~ exp(-x)/x^2*(1 - 3/x + 11/x^2 - 50/x^3 + 274/x^4 - 1764/x^5 + 13068/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 number of permutations of [n+1] containing exactly 2 cycles. Example: a(2) = 3 because the permutations (1)(23), (12)(3), (13)(2) are the only permutations of [3] with exactly 2 cycles. - Tom Woodward (twoodward(AT)macalester.edu), Nov 12 2009
It appears that, with the exception of n= 4, a(n) mod n = 0 if n is composite and = n-1 if n is prime. - Gary Detlefs, Sep 11 2010
a(n) is a multiple of A025527(n). - Charles R Greathouse IV, Oct 16 2012
Numerator of harmonic number H(n) = Sum_{i=1..n} 1/i when not reduced. See A001008 (Wolstenholme numbers) for the reduced numerators. - Rahul Jha, Feb 18 2015
The Stirling transform of this sequence is A222058(n) (Harmonic-geometric numbers). - Anton Zakharov, Aug 07 2016
a(n) is the (n-1)-st elementary symmetric function of the first n numbers. - Anton Zakharov, Nov 02 2016
The n-th iterated integral of log(x) is x^n * (n! * log(x) - a(n))/(n!)^2 + a polynomial of degree n-1 with arbitrary coefficients. This can be proven using the recurrence relation a(n) = (n-1)! + n*a(n-1). - Mohsen Maesumi, Oct 31 2018
Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019
Total number of left-to-right maxima (or minima) in all permutations of [n]. a(3) = 11 = 3+2+2+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21. - Alois P. Heinz, Aug 01 2020

Examples

			(1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...
G.f. = x + x^2 + 5*x^3 + 14*x^4 + 94*x^5 + 444*x^6 + 3828*x^7 + 25584*x^8 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identities 186-190.
  • N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, 1986, see page 2. MR0863284 (89d:41049)
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
  • Shanzhen Gao, Permutations with Restricted Structure (in preparation).
  • K. Javorszky, Natural Orders: De Ordinibus Naturalibus, 2016, ISBN 978-3-99057-139-2.
  • 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).

Links

Crossrefs

Cf. A000399, A000454, A000482, A001233, A001234, A243569, A243570.
Cf. A000774, A004041, A024167, A046674, A049034, A008275.
Cf. A021009, A081358, A092691, A094587, A151881, A121633.
With signs: A081048.
Column 1 in triangle A008969.
Row sums of A136662.

Programs

  • Magma
    a:=[]; for n in [1..22] do a:=a cat [Abs(StirlingFirst(n,2))]; end for; a; // Marius A. Burtea, Jan 01 2020
  • Maple
    A000254 := proc(n) option remember; if n<=1 then n else n*A000254(n-1)+(n-1)!; fi; end: seq(A000254(n),n=0..21);
a := n -> add(n!/k, k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jan 22 2008
  • Mathematica
    Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] (* Wouter Meeussen *)
Table[ n!*HarmonicNumber[n], {n, 0, 19}] (* Robert G. Wilson v, May 21 2005 *)
Table[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}],{n,1,30}] (* Alexander Adamchuk, Jul 11 2006 *)
Abs[StirlingS1[Range[20],2]] (* Harvey P. Dale, Aug 16 2011 *)
Table[Gamma'[n + 1] /. EulerGamma -> 0, {n, 0, 30}] (* Li Han, Feb 14 2024*)
  • Maxima
    a(n):=(-1)^(n+1)/2*(n+1)*sum(k*bern(k-1)*stirling1(n,k),k,1,n); /* Vladimir Kruchinin, Nov 20 2016 */
  • MuPAD
    A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) := 1:
  • PARI
    {a(n) = if( n<0, 0, (n+1)! / 2 * sum( k=1, n, 1 / k / (n+1-k)))} /* Michael Somos, Feb 05 2004 */
  • Sage
    [stirling_number1(i, 2) for i in range(1, 22)]  # Zerinvary Lajos, Jun 27 2008

Formula

Let P(n,X) = (X+1)*(X+2)*(X+3)*...*(X+n); then a(n) is the coefficient of X; or a(n) = P'(n,0). - Benoit Cloitre, May 09 2002
Sum_{k > 0} a(k) * x^k/ k!^2 = exp(x) *(Sum_{k>0} (-1)^(k+1) * x^k / (k * k!)). - Michael Somos, Mar 24 2004; corrected by Warren D. Smith, Feb 12 2006
a(n) is the coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/2.
a(n) = n! * Sum_{i=1..n} 1/i = n! * H(n), where H(n) = A001008(n)/A002805(n) is the n-th harmonic number.
a(n) ~ 2^(1/2)*Pi^(1/2)*log(n)*n^(1/2)*e^-n*n^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: log(1 - x) / (x-1). (= (log(1 - x))^2 / 2 if offset 1). - Michael Somos, Feb 05 2004
D-finite with recurrence: a(n) = a(n-1) * (2*n - 1) - a(n-2) * (n - 1)^2, if n > 1. - Michael Somos, Mar 24 2004
a(n) = A081358(n)+A092691(n). - Emeric Deutsch, Aug 12 2004
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic, Jan 29 2005
p^2 divides a(p-1) for prime p > 3. a(n) = (Sum_{i=1..n} 1/i) / Product_{i=1..n} 1/i. - Alexander Adamchuk, Jul 11 2006
a(n) = 3* A001710(n) + 2* A001711(n-3) for n > 2; e.g., 11 = 3*3 + 2*1, 50 = 3*12 + 2*7, 274 = 3*60 + 2*47, ... - Gary Detlefs, May 24 2010
a(n) = A138772(n+1) - A159324(n). - Gary Detlefs, Jul 05 2010
a(n) = A121633(n) + A002672(n). - Gary Detlefs, Jul 18 2010
a(n+1) = Sum_{i=1..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 14 2010
From Gary Detlefs, Sep 11 2010: (Start)
a(n) = (a(n-1)*(n^2 - 2*n + 1) + (n + 1)!)/(n - 1) for n > 2.
It appears that, with the exception of n = 2, (a(n+1)^2 - a(n)^2) mod n^2 = 0 if n is composite and 4*n if n is prime.
It appears that, with the exception of n = 2, (a(n+1)^3 - a(n)^2) mod n = 0 if n is composite and n - 2 if n is prime.
It appears that, with the exception of n = 2, (a(n)^2 + a(n+1)^2) mod n = 0 if n is composite and = 2 if n is prime. (End)
a(n) = Integral_{x=0..oo} (x^n - n!)*log(x)*exp(-x) dx. - Groux Roland, Mar 28 2011
a(n) = 3*n!/2 + 2*(n-2)!*Sum_{k=0..n-3} binomial(k+2,2)/(n-2-k) for n >= 2. - Gary Detlefs, Sep 02 2011
a(n)/(n-1)! = ml(n) = n*ml(n-1)/(n-1) + 1 for n > 1, where ml(n) is the average number of random draws from an n-set with replacement until the total set has been observed. G.f. of ml: x*(1 - log(1 - x))/(1 - x)^2. - Paul Weisenhorn, Nov 18 2011
a(n) = det(|S(i+2, j+1)|, 1 <= i,j <= n-2), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
E.g.f.: x/(1 - x)*E(0)/2, where E(k) = 2 + E(k+1)*x*(k + 1)/(k + 2). - Sergei N. Gladkovskii, Jun 01 2013 [Edited by Michael Somos, Nov 28 2013]
0 = a(n) * (a(n+4) - 6*a(n+3) + 7*a(n+2) - a(n+1)) - a(n+1) * (4*a(n+3) - 6*a(n+2) + a(n+1)) + 3*a(n+2)^2 unless n=0. - Michael Somos, Nov 28 2013
For a simple way to calculate the sequence, multiply n! by the integral from 0 to 1 of (1 - x^n)/(1 - x) dx. - Rahul Jha, Feb 18 2015
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Inverse binomial transform of A073596.
a(n) ~ sqrt(2*Pi*n) * n^n * (log(n) + gamma)/exp(n), where gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = ((-1)^(n+1)/2*(n+1))*Sum_{k=1..n} k*Bernoulli(k-1)*Stirling1(n,k). - Vladimir Kruchinin, Nov 20 2016
a(n) = (n)! * (digamma(n+1) + gamma), where gamma is the Euler-Mascheroni constant A001620. - Pedro Caceres, Mar 10 2018
From Andy Nicol, Oct 21 2021: (Start)
Gamma'(x) = a(x-1) - (x-1)!*gamma, where Gamma'(x) is the derivative of the gamma function at positive integers and gamma is the Euler-Mascheroni constant. E.g.:
Gamma'(1) = -gamma, Gamma'(2) = 1-gamma, Gamma'(3) = 3-2*gamma,
Gamma'(22) = 186244810780170240000 - 51090942171709440000*gamma. (End)
From Peter Bala, Feb 03 2022: (Start)
The following are all conjectural:
E.g.f.: for nonzero m, (1/m)*Sum_{n >= 1} (-1)^(n+1)*(1/n)*binomial(m*n,n)* x^n/(1 - x)^(m*n+1) = x + 3*x^2/2! + 11*x^3/3! + 50*x^4/4! + ....
For nonzero m, a(n) = (1/m)*n!*Sum_{k = 1..n} (-1)^(k+1)*(1/k)*binomial(m*k,k)* binomial(n+(m-1)*k,n-k).
a(n)^2 = (1/2)*n!^2*Sum_{k = 1..n} (-1)^(k+1)*(1/k^2)*binomial(n,k)* binomial(n+k,k). (End)
From Mélika Tebni, Jun 20 2022: (Start)
a(n) = -Sum_{k=0..n} k!*A021009(n, k+1).
a(n) = Sum_{k=0..n} k!*A094587(n, k+1). (End)
a(n) = n! * 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n - 1)^2/((2*n - 1)))))). - Peter Bala, Mar 16 2024

A269951 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 5, 1, 0, 16, 23, 9, 1, 0, 65, 116, 65, 14, 1, 0, 326, 669, 470, 145, 20, 1, 0, 1957, 4429, 3634, 1415, 280, 27, 1, 0, 13700, 33375, 30681, 14084, 3535, 490, 35, 1, 0, 109601, 283072, 284066, 147532, 43939, 7756, 798, 44, 1
Offset: 0

Views

Author

Peter Luschny, Apr 10 2016

Keywords

Examples

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   5,   5,   1;
  0,  16,  23,   9,   1;
  0,  65, 116,  65,  14,  1;
  0, 326, 669, 470, 145, 20, 1;

Links

Crossrefs

A000522 (col. 1), A073596 (col. 2), A000096 (diag. n-1), A241765 (diag. n-2).
A001339 (row sums), A137597 (unsigned matrix inverse).
Cf. A132393, A269952.

Programs

  • Maple
    A269951 := (n,k) -> add((-1)^(n-j)*binomial(-j,-n)*abs(Stirling1(j,k)), j=0..n):
seq(seq(A269951(n,k), k=0..n), n=0..9);
  • Mathematica
    Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] Abs[StirlingS1[j,k]], {j,0,n}], {n,0,9}, {k,0,n}]]

A300490 Expansion of e.g.f. -exp(-x)*log(1 - x)/(1 - x).

Original entry on oeis.org

0, 1, 1, 5, 20, 109, 689, 5053, 42048, 391641, 4036697, 45618341, 560889988, 7454314789, 106488455033, 1627269878557, 26487441519584, 457532446622001, 8359188190686609, 161056273132588933, 3263644496701880404, 69389030027882288861, 1544501472271318499105
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

Comments

Inverse binomial transform A000254.

Examples

			-exp(-x)*log(1 - x)/(1 - x) = x/1! + x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + 689*x^6/6! + 5053*x^7/7! + ...

Links

Crossrefs

Cf. A000254, A073596.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<2, n, n*b(n-1)+(n-1)!) end:
a:= proc(n) add(b(k)*(-1)^(n-k)*binomial(n, k), k=0..n) end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 07 2018
  • Mathematica
    nmax = 22; CoefficientList[Series[-Exp[-x] Log[1 - x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n,k] k! HarmonicNumber[k], {k, 1, n}], {n, 0, 22}]
a[0] = 0; a[1] = a[2] = 1; a[3] = 5; a[n_Integer] := a[n] = (2 n - 3) a[n - 1] - (n^2 - 6 n + 7) a[n - 2] - (n - 2) (2 n - 5) a[n - 3] - (n - 3) (n - 2) a[n - 4]; Table[a[n], {n, 0, 22}] (* Vladimir Reshetnikov, Mar 30 2022 *)

Formula

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*k!*H(k), where H(k) is the k-th harmonic number.
a(n) ~ n! * (log(n) + gamma) / exp(1), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 23 2018
Recurrence: a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 5, a(n+4) = (2*n+5)*a(n+3) - (n^2+2*n-1)*a(n+2) - (2*n+3)*(n+2)*a(n+1) - (n+2)*(n+1)*a(n). - Vladimir Reshetnikov, Mar 30 2022

A302547 Expansion of e.g.f. -log(1 - log(1 + x))/(1 - log(1 + x)).

Original entry on oeis.org

0, 1, 2, 4, 11, 33, 131, 516, 2810, 12934, 97870, 447940, 5308112, 16394116, 450505844, -315178912, 60774618672, -394330113648, 12662225550288, -157622647720032, 3766647294946944, -64679214198647520, 1475157821754785184, -30431206030329719424, 719032203373502252160
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 4*x^3/3! + 11*x^4/4! + 33*x^5/5! + 131*x^6/6! + ...

Links

Crossrefs

Cf. A000254, A001008, A002805, A006252, A073596, A089064, A222058, A300490, A302548.

Programs

  • Maple
    H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(Stirling1(n, k)*H(k)*k!, k=1..n):
seq(a(n), n=0..27);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - Log[1 + x]]/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 24}]

Formula

a(n) = Sum_{k=1..n} Stirling1(n,k)*H(k)*k!, where H(k) is the k-th harmonic number.

A302548 Expansion of e.g.f. -log(1 + log(1 - x))/(1 + log(1 - x)).

Original entry on oeis.org

0, 1, 4, 22, 155, 1333, 13541, 158688, 2107682, 31291894, 513590170, 9234669420, 180534475832, 3812852144788, 86517295628188, 2099170738243328, 54233876338638192, 1486517654443664016, 43084555863325589232, 1316588795487600071904, 42306543064537291007424, 1426115146736949130634400
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 22*x^3/3! + 155*x^4/4! + 1333*x^5/5! + 13541*x^6/6! + ...

Links

Crossrefs

Cf. A000254, A001008, A002805, A003713, A007840, A073596, A222058, A300490, A302547.

Programs

  • Maple
    H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*H(k)*k!, k=1..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[-Log[1 + Log[1 - x]]/(1 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 21}]

Formula

a(n) = Sum_{k=1..n} |Stirling1(n,k)|*H(k)*k!, where H(k) is the k-th harmonic number.
a(n) ~ sqrt(2*Pi) * log(n) * n^(n + 1/2) / (exp(1)-1)^(n+1). - Vaclav Kotesovec, Jun 23 2018

A381021 Expansion of e.g.f. log(1-x)^2 * exp(x) / 2.

Original entry on oeis.org

0, 0, 1, 6, 29, 145, 814, 5243, 38618, 321690, 2995011, 30840304, 348114711, 4274888891, 56744495872, 809667333733, 12358833406580, 200955441549140, 3467781770502885, 63298198354605210, 1218507112218768721, 24671782054230662277, 524152965820457130290
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2025

Keywords

Crossrefs

Column k=2 of A094816.
Cf. A073596.

Programs

  • Mathematica
    nmax=22; CoefficientList[Series[Log[1-x]^2*Exp[x]/2,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 12 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(k, 2, 1)));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * |Stirling1(k,2)|.

A381107 Expansion of e.g.f. -log(1-x) * (exp(x) - 1) / (1-x).

Original entry on oeis.org

0, 0, 2, 12, 66, 395, 2665, 20307, 173488, 1646745, 17216653, 196730567, 2440331300, 32666847941, 469457190501, 7210003071247, 117862325748960, 2043420738374545, 37453428525580725, 723643767046525111, 14700326905250293556, 313236372986056228013, 6985951253209713959645
Offset: 0

Views

Author

Seiichi Manyama, Feb 14 2025

Keywords

Crossrefs

Cf. A002627, A381108.
Cf. A000254, A073596.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(n, k)*abs(stirling(k+1, 2, 1)));

Formula

a(n) = Sum_{k=0..n-1} binomial(n,k) * |Stirling1(k+1,2)|.
a(n) = A073596(n) - A000254(n).

A300488 a(n) = n! * [x^n] -exp(n*x)*log(1 - x)/(1 - x).

Original entry on oeis.org

0, 1, 7, 65, 770, 11149, 191124, 3788469, 85281552, 2149582761, 59983774240, 1835925702137, 61157508893568, 2202760340194517, 85303050939131648, 3534478528925155725, 156026612737389987840, 7310587974761946511761, 362356607517279564386304, 18943214212273585171456753
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of e.g.f. -exp(n*x)*log(1 - x)/(1 - x) begins:
n = 0: (0), 1,   3,   11,    50,     274,  ...
n = 1:  0, (1),  5,   23,   116,     669,  ...
n = 2:  0,  1,  (7),  41,   242,    1534,  ...
n = 3:  0,  1,   9,  (65),  452,    3229,  ...
n = 4:  0,  1,  11,   95,  (770),   6234,  ...
n = 5:  0,  1,  13,  131,  1220,  (11149), ...
...
This sequence is the main diagonal of the table.

Links

Crossrefs

Cf. A000254, A065456, A073596.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[-Exp[n x] Log[1 - x]/(1 - x), {x, 0, n}], {n, 0, 19}]
Table[Sum[n^(n - k) Binomial[n,k] k! HarmonicNumber[k], {k, 1, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=1..n} n^(n-k)*binomial(n,k)*k!*H(k), where H(k) is the k-th harmonic number.
Showing 1-8 of 8 results.

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