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 1-5 of 5 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

Views

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

A004041 Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).

Original entry on oeis.org

1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, 15188465029114325025, 537928935889764226500
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Comments

n-th elementary symmetric function of the first n+1 odd positive integers.
Also the determinant of the n X n matrix given by m(i,j) = 2*i + 2 = if i = j and otherwise 1. For example, Det[{{4, 1, 1, 1, 1, 1}, {1, 6, 1, 1, 1, 1}, {1, 1, 8, 1, 1, 1}, {1, 1, 1, 10, 1, 1}, {1, 1, 1, 1, 12, 1}, {1, 1, 1, 1, 1, 14}}] = 264207 = a(6). - John M. Campbell, May 20 2011

Examples

			(arctanh(x))^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...

Links

Crossrefs

Cf. A000254, A024199, A049034.
Cf. A002428.
From Johannes W. Meijer, Jun 08 2009: (Start)
Equals second left hand column of A028338 triangle.
Equals second right hand column of A109692 triangle.
Equals second left hand column of A161198 triangle divided by 2.
(End)

Programs

  • Mathematica
    Table[(-1)^(n + 1)* Sum[(-2)^(n - k) k (-1)^(n - k) StirlingS1[n + 1, k + 1], {k, 0, n}], {n, 1, 18}] (* Zerinvary Lajos, Jul 08 2009 *)
FunctionExpand@Table[(2 n + 1)!! (Log[4] + HarmonicNumber[n + 1/2])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 13 2016 *)

Formula

a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).
a(n) is coefficient of x^(2*n+2) in (arctanh x)^2, multiplied by (n + 1)*(2*n + 1)!!.
a(n) = Sum_{i=k+1..n} (-1)^(k+1-i)*2^(n-1)*binomial(i-1, k)*s1(n, i) with k = 1, where s1(n, i) are unsigned Stirling numbers of the first kind. - Victor Adamchik (adamchik(AT)ux10.sp.cs.cmu.edu), Jan 23 2001
a(n) ~ 2^(1/2)*log(n)*n*(2n/e)^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: 1/2*(1 - 2*x)^(-3/2)*(2 - log(1 - 2*x)). - Vladeta Jovovic, Feb 19 2003
Sum_{n>=1} a(n-1)/(n!*n*2^n) = (Pi/2)^2. - Philippe Deléham, Aug 12 2003
For n >= 1, a(n-1) = 2^(n-1)*n!*(Sum_{k=0..n-1} (-1)^k*binomial(1/2, k)/(n - k)). - Milan Janjic, Dec 14 2008
Recurrence: a(n) = 4*n*a(n-1) - (2*n - 1)^2*a(n-2). - Vladimir Reshetnikov, Oct 13 2016

A081358 E.g.f.: log((1+x) / (1-x)) / (2*(1-x)).

Original entry on oeis.org

0, 1, 2, 8, 32, 184, 1104, 8448, 67584, 648576, 6485760, 74972160, 899665920, 12174658560, 170445219840, 2643856588800, 42301705420800, 740051782041600, 13320932076748800, 259500083163955200, 5190001663279104000, 111422936937037824000, 2451304612614832128000
Offset: 0

Views

Author

Michael Somos, Mar 18 2003

Keywords

Comments

Number of cycles of odd cardinality in all permutations of [n]. Example: a(3)=8 because among (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) we have eight cycles of odd length. - Emeric Deutsch, Aug 12 2004
a(n) is a function of the harmonic numbers. a(n) = n!*h(n) - n!/2 * h(floor(n/2)), where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Aug 06 2010

Examples

			G.f. = x + 2*x^2 + 8*x^3 + 32*x^4 + 184*x^5 + 1104*x^6 + 8448*x^7 + ...

References

  • I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 3.3.13

Links

Crossrefs

Cf. A000254, A001640, A049034, A092691, A151884.

Programs

  • Mathematica
    nn = 20; Range[0, nn]! CoefficientList[
  D[Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], y] /. y -> 1, x]  (* Geoffrey Critzer, Aug 27 2012 *)
a[ n_] := If[ n < 0, 0, n! Sum[ 1/k, {k, 1, n, 2}]]; (* Michael Somos, Jan 06 2015 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ (1 + x) / (1 - x)] / (2 (1 - x)), {x, 0, n}]]; (* Michael Somos, Jan 06 2015 *)
  • PARI
    {a(n) = if( n<1, 0, n! * polcoeff( log(1 + 2 / (-1 + 1 / (x + x * O(x^n)))) / (2 * (1-x)), n))};
  • PARI
    {a(n) = if( n<0, 0, n! * sum(k=1, n, (k%2)/k))}; /* Michael Somos, Sep 19 2006 */
  • PARI
    first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017

Formula

E.g.f.: log((1+x) / (1-x)) / (2*(1-x)).
a(n) = n! * Sum_{k=0..n, k odd} 1/k.
a(n) = n!/2*(Psi(ceiling(n/2)+1/2)+gamma+2*log(2)). - Vladeta Jovovic, Oct 20 2003
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*2^(k-1)*binomial(n, k)/k. - Vladeta Jovovic, Aug 12 2005
a(n) = n*a(n-1) + ((-1)^(n+1)+1)/2*(n-1)!. - Gary Detlefs, Aug 06 2010
a(n) = A000254(n) - A092691(n). - Gary Detlefs, Aug 06 2010
a(n) ~ n!/2 * (log(n) + gamma + log(2)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 05 2013
a(2*n + 1) = A049034(n).
E.g.f.: arctanh(x)/(1 - x). - Ilya Gutkovskiy, Dec 19 2017

A046674 a(n) = A046673(n)/2.

Original entry on oeis.org

1, 18, 660, 42000, 4142880, 586776960, 113020427520, 28432576972800, 9056055981772800, 3562946373482496000, 1697172166720622592000, 962691052768668647424000, 641260394342965606809600000, 495681727573700350908825600000, 440084864982859007245221888000000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A000254, A049034, A046673.

Extensions

Offset changed to 1 by Jon E. Schoenfield, Mar 08 2020

A008309 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

Original entry on oeis.org

1, 1, -2, 1, -8, 1, 24, -20, 1, 184, -40, 1, -720, 784, -70, 1, -8448, 2464, -112, 1, 40320, -52352, 6384, -168, 1, 648576, -229760, 14448, -240, 1, -3628800, 5360256, -804320, 29568, -330, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			With the zero coefficients included the data begins 1; 0,1; -2,0,1; 0,-8,0,1; 24,0,-20,0,1; 0,184,0,-40,0,1; ..., which is A049218.
The table without zeros begins
    1;
    1;
   -2,   1;
   -8,   1;
   24, -20,   1;
  184, -40,   1;
  ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.

Crossrefs

Essentially same as A049218.
A007290(n) = -T(n, floor(n-1)/2);
A010050(n) = (-1)^n*T(2n+1, 1);
A049034(n) = (-1)^n*T(2n+2, 1);
A049214(n) = (-1)^n*T(2n+3, 2);
A049215(n) = (-1)^n*T(2n+4, 2);
A049216(n) = (-1)^n*T(2n+5, 3);
A049217(n) = (-1)^n*T(2n+6, 3).

Programs

  • Mathematica
    t[n_, k_] := (-1)^((3*n+k)/2)*n!/2^k*Sum[2^i*Binomial[n-1, i-1]*StirlingS1[i, k]/i!, {i, k, n}]; Flatten[Table[t[n,k], {n,1,11}, {k, 2-Mod[n, 2], n, 2}]] (* Jean-François Alcover, Aug 31 2011, after Vladimir Kruchinin *)
  • PARI
    a(n)=atan(x)^n/n!
T(n,k)=polcoeff(serlaplace(a(2*k-n%2)), n)

Formula

E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(m, floor((k+1)/2))* x^m/m!, where m = 2*n + k mod 2.

Extensions

Additional comments from Michael Somos
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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