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.

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A001563 a(n) = n*n! = (n+1)! - n!.

Original entry on oeis.org

0, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A similar sequence, with the initial 0 replaced by 1, namely A094258, is defined by the recurrence a(2) = 1, a(n) = a(n-1)*(n-1)^2/(n-2). - Andrey Ryshevich (ryshevich(AT)notes.idlab.net), May 21 2002
Denominators in power series expansion of E_1(x) + gamma + log(x), x > 0. - Michael Somos, Dec 11 2002
If all the permutations of any length k are arranged in lexicographic order, the n-th term in this sequence (n <= k) gives the index of the permutation that rotates the last n elements one position to the right. E.g., there are 24 permutations of 4 items. In lexicographic order they are (0,1,2,3), (0,1,3,2), (0,2,1,3), ... (3,2,0,1), (3,2,1,0). Permutation 0 is (0,1,2,3), which rotates the last 1 element, i.e., it makes no change. Permutation 1 is (0,1,3,2), which rotates the last 2 elements. Permutation 4 is (0,3,1,2), which rotates the last 3 elements. Permutation 18 is (3,0,1,2), which rotates the last 4 elements. The same numbers work for permutations of any length. - Henry H. Rich (glasss(AT)bellsouth.net), Sep 27 2003
Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,154,...]. - Michael Somos, Mar 04 2004
From Michael Somos, Apr 27 2012: (Start)
Stirling transform of a(n)=[1,4,18,96,...] is A069321(n)=[1,5,31,233,...].
Partial sums of a(n)=[0,1,4,18,...] is A033312(n+1)=[0,1,5,23,...].
Binomial transform of A000166(n+1)=[0,1,2,9,...] is a(n)=[0,1,4,18,...].
Binomial transform of A000255(n+1)=[1,3,11,53,...] is a(n+1)=[1,4,18,96,...].
Binomial transform of a(n)=[0,1,4,18,...] is A093964(n)=[0,1,6,33,...].
Partial sums of A001564(n)=[1,3,4,14,...] is a(n+1)=[1,4,18,96,...].
(End)
Number of small descents in all permutations of [n+1]. A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) =1. Example: a(2)=4 because there are 4 small descents in the permutations 123, 13\2, 2\13, 231, 312, 3\2\1 of {1,2,3} (shown by \). a(n)=Sum_{k=0..n-1}k*A123513(n,k). - Emeric Deutsch, Oct 02 2006
Equivalently, in the notation of David, Kendall and Barton, p. 263, this is the total number of consecutive ascending pairs in all permutations on n+1 letters (cf. A010027). - N. J. A. Sloane, Apr 12 2014
a(n-1) is the number of permutations of n in which n is not fixed; equivalently, the number of permutations of the positive integers in which n is the largest element that is not fixed. - Franklin T. Adams-Watters, Nov 29 2006
Number of factors in a determinant when writing down all multiplication permutations. - Mats Granvik, Sep 12 2008
a(n) is also the sum of the positions of the left-to-right maxima in all permutations of [n]. Example: a(3)=18 because the positions of the left-to-right maxima in the permutations 123,132,213,231,312 and 321 of [3] are 123, 12, 13, 12, 1 and 1, respectively and 1+2+3+1+2+1+3+1+2+1+1=18. - Emeric Deutsch, Sep 21 2008
Equals eigensequence of triangle A002024 ("n appears n times"). - Gary W. Adamson, Dec 29 2008
Preface the series with another 1: (1, 1, 4, 18, ...); then the next term = dot product of the latter with "n occurs n times". Example: 96 = (1, 1, 4, 8) dot (4, 4, 4, 4) = (4 + 4 + 16 + 72). - Gary W. Adamson, Apr 17 2009
Row lengths of the triangle in A030298. - Reinhard Zumkeller, Mar 29 2012
a(n) is also the number of minimum (n-)distinguishing labelings of the star graph S_{n+1} on n+1 nodes. - Eric W. Weisstein, Oct 14 2014
When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the right, i.e., a(n) is the permutation with the cycle notation (0 1 ... n-1 n). Compare array A051683 for circular shifts to the right in a broader sense. Compare sequence A007489 for circular shifts to the left. - Tilman Piesk, Apr 29 2017
a(n-1) is the number of permutations on n elements with no cycles of length n. - Dennis P. Walsh, Oct 02 2017
The number of pandigital numbers in base n+1, such that each digit appears exactly once. For example, there are a(9) = 9*9! = 3265920 pandigital numbers in base 10 (A050278). - Amiram Eldar, Apr 13 2020

Examples

			E_1(x) + gamma + log(x) = x/1 - x^2/4 + x^3/18 - x^4/96 + ..., x > 0. - _Michael Somos_, Dec 11 2002
G.f. = x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 4320*x^6 + 35280*x^7 + 322560*x^8 + ...

References

  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 218.
  • J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.
  • F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
  • 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).
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 37, equation 37:6:1 at page 354.

Links

Crossrefs

Cf. A000142, A002024, A047920, A047922, A053495, A055089, A123513, A143946, A229837, A239069.
Cf. A163931 (E(x,m,n)), A002775 (n^2*n!), A091363 (n^3*n!), A091364 (n^4*n!).
Cf. sequences with formula (n + k)*n! listed in A282466.
Row sums of A185105, A322383, A322384, A094485.

Programs

  • GAP
    List([0..20], n-> n*Factorial(n) ); # G. C. Greubel, Dec 30 2019
  • Haskell
    a001563 n = a001563_list !! n
a001563_list = zipWith (-) (tail a000142_list) a000142_list
-- Reinhard Zumkeller, Aug 05 2013
  • Magma
    [Factorial(n+1)-Factorial(n): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014
  • Maple
    A001563 := n->n*n!;
  • Mathematica
    Table[n!n,{n,0,25}] (* Harvey P. Dale, Oct 03 2011 *)
  • PARI
    {a(n) = if( n<0, 0, n * n!)} /* Michael Somos, Dec 11 2002 */
  • Sage
    [n*factorial(n) for n in (0..20)] # G. C. Greubel, Dec 30 2019

Formula

From Michael Somos, Dec 11 2002: (Start)
E.g.f.: x / (1 - x)^2.
a(n) = -A021009(n, 1), n >= 0. (End)
The coefficient of y^(n-1) in expansion of (y+n!)^n, n >= 1, gives the sequence 1, 4, 18, 96, 600, 4320, 35280, ... - Artur Jasinski, Oct 22 2007
Integral representation as n-th moment of a function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(x*(x-1)*exp(-x)) dx, for n>=0. This representation may not be unique. - Karol A. Penson, Sep 27 2001
a(0)=0, a(n) = n*a(n-1) + n!. - Benoit Cloitre, Feb 16 2003
a(0) = 0, a(n) = (n - 1) * (1 + Sum_{i=1..n-1} a(i)) for i > 0. - Gerald McGarvey, Jun 11 2004
Arises in the denominators of the following identities: Sum_{n>=1} 1/(n*(n+1)*(n+2)) = 1/4, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)) = 1/18, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)*(n+4)) = 1/96, etc. The general expression is Sum_{n>=k} 1/C(n, k) = k/(k-1). - Dick Boland, Jun 06 2005 [And the general expression implies that Sum_{n>=1} 1/(n*(n+1)*...*(n+k-1)) = (Sum_{n>=k} 1/C(n, k))/k! = 1/((k-1)*(k-1)!) = 1/a(k-1), k >= 2. - Jianing Song, May 07 2023]
a(n) = Sum_{m=2..n+1} |Stirling1(n+1, m)|, n >= 1 and a(0):=0, where Stirling1(n, m) = A048994(n, m), n >= m = 0.
a(n) = 1/(Sum_{k>=0} k!/(n+k+1)!), n > 0. - Vladeta Jovovic, Sep 13 2006
a(n) = Sum_{k=1..n(n+1)/2} k*A143946(n,k). - Emeric Deutsch, Sep 21 2008
The reciprocals of a(n) are the lead coefficients in the factored form of the polynomials obtained by summing the binomial coefficients with a fixed lower term up to n as the upper term, divided by the term index, for n >= 1: Sum_{k = i..n} C(k, i)/k = (1/a(n))*n*(n-1)*..*(n-i+1). The first few such polynomials are Sum_{k = 1..n} C(k, 1)/k = (1/1)*n, Sum_{k = 2..n} C(k, 2)/k = (1/4)*n*(n-1), Sum_{k = 3..n} C(k, 3)/k = (1/18)*n*(n-1)*(n-2), Sum_{k = 4..n} C(k, 4)/k = (1/96)*n*(n-1)*(n-2)*(n-3), etc. - Peter Breznay (breznayp(AT)uwgb.edu), Sep 28 2008
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) then a(n) = (-1)^(n-1)*f(n,1,-2), (n >= 1). - Milan Janjic, Mar 01 2009
Sum_{n>=1} (-1)^(n+1)/a(n) = 0.796599599... [Jolley eq. 289]
G.f.: 2*x*Q(0), where Q(k) = 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: W(0)*(1-sqrt(x)) - 1, where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+2)/(sqrt(x)*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 18 2013
G.f.: T(0)/x - 1/x, where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013
G.f.: Q(0)*(1-x)/x - 1/x, where Q(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
D-finite with recurrence: a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, Jan 14 2020
a(n) = (-1)^(n+1)*(n+1)*Sum_{k=1..n} A094485(n,k)*Bernoulli(k). The inverse of the Worpitzky representation of the Bernoulli numbers. - Peter Luschny, May 28 2020
From Amiram Eldar, Aug 04 2020: (Start)
Sum_{n>=1} 1/a(n) = Ei(1) - gamma = A229837.
Sum_{n>=1} (-1)^(n+1)/a(n) = gamma - Ei(-1) = A239069. (End)
a(n) = Gamma(n)*A000290(n) for n > 0. - Jacob Szlachetka, Jan 01 2022

A001048 a(n) = n! + (n-1)!.

Original entry on oeis.org

2, 3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680, 43545600, 518918400, 6706022400, 93405312000, 1394852659200, 22230464256000, 376610217984000, 6758061133824000, 128047474114560000, 2554547108585472000, 53523844179886080000, 1175091669949317120000
Offset: 1

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

Number of {12, 12*, 1*2, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
a(n) is the hook product of the shape (n, 1). - Emeric Deutsch, May 13 2004
From Jaume Oliver Lafont, Dec 01 2009: (Start)
(1+(x-1)*exp(x))/x = Sum_{k >= 1} x^k/a(k).
Setting x = 1 yields Sum_{k >= 1} 1/a(k) = 1. [Jolley eq 302] (End)
With regard to the comment by Jaume Oliver Lafont: P(n) = 1/a(n) is a probability distribution, with all values given as unit fractions. This distribution is connected to the Irwin-Hall distribution: Consider successively drawn random numbers, uniformly distributed in [0,1]. 1/a(n) is the probability for the sum of the random numbers exceeding 1 exactly with the (n+1)-th summand. P(n) has mean e-1 and variance 3e-e^2. From this we get e as the expected number of summands. - Manfred Boergens, May 20 2024
For n >= 2, a(n) is the size of the largest conjugacy class of the symmetric group on n + 1 letters. Equivalently, the maximum entry in each row of A036039. - Geoffrey Critzer, May 19 2013
In factorial base representation (A007623) the terms are written as: 10, 11, 110, 1100, 11000, 110000, ... From a(2) = 3 = "11" onward each term begins always with two 1's, followed by n-2 zeros. - Antti Karttunen, Sep 24 2016
e is approximately a(n)/A000255(n-1) for large n. - Dale Gerdemann, Jul 26 2019
a(n) is the number of permutations of [n+1] in which all the elements of [n] are cycle-mates, that is, 1,..,n are all in the same cycle. This result is readily shown after noting that the elements of [n] can be members of a n-cycle or an (n+1)-cycle. Hence a(n)=(n-1)!+n!. See an example below. - Dennis P. Walsh, May 24 2020

Examples

			For n=3, a(3) counts the 8 permutations of [4] with 1,2, and 3 all in the same cycle, namely, (1 2 3)(4), (1 3 2)(4), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 2 4 3), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961.
  • 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

Apart from initial terms, same as A059171.
Equals the square root of the first right hand column of A162990. - Johannes W. Meijer, Jul 21 2009
From a(2)=3 onward the second topmost row of arrays A276588 and A276955.
Cf. sequences with formula (n + k)*n! listed in A282466, A334397.

Programs

Formula

a(n) = (n+1)*(n-1)!.
E.g.f.: x/(1-x) - log(1-x). - Ralf Stephan, Apr 11 2004
The sequence 1, 3, 8, ... has g.f. (1+x-x^2)/(1-x)^2 and a(n) = n!(n + 2 - 0^n) = n!A065475(n) (offset 0). - Paul Barry, May 14 2004
a(n) = (n+1)!/n. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003
Factorial expansion of 1: 1 = sum_{n > 0} 1/a(n) [Jolley eq 302]. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003
a(1) = 2, a(2) = 3, D-finite recurrence a(n) = (n^2 - n - 2)*a(n-2) for n >= 3. - Jaume Oliver Lafont, Dec 01 2009
a(n) = ((n+2)A052649(n) - A052649(n+1))/2. - Gary Detlefs, Dec 16 2009
G.f.: U(0) where U(k) = 1 + (k+1)/(1 - x/(x + 1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 25 2012
G.f.: 2*(1+x)/x/G(0) - 1/x, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
a(n) = (n-1)*a(n-1) + (n-1)!. - Bruno Berselli, Feb 22 2017
a(1)=2, a(2)=3, D-finite recurrence a(n) = (n-1)*a(n-1) + (n-2)*a(n-2). - Dale Gerdemann, Jul 26 2019
a(n) = 2*A000255(n-1) + A096654(n-2). - Dale Gerdemann, Jul 26 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - 2/e (A334397). - Amiram Eldar, Jan 13 2021

Extensions

More terms from James Sellers, Sep 19 2000

A062119 a(n) = n! * (n-1).

Original entry on oeis.org

0, 2, 12, 72, 480, 3600, 30240, 282240, 2903040, 32659200, 399168000, 5269017600, 74724249600, 1133317785600, 18307441152000, 313841848320000, 5690998849536000, 108840352997376000, 2189611807358976000, 46225138155356160000, 1021818843434188800000
Offset: 1

Views

Author

Olivier Gérard, Jun 13 2001

Keywords

Comments

For n > 0, a(n) = number of permutations of length n+1 that have 2 predetermined elements nonadjacent; e.g., for n=2, the permutations with, say, 1 and 2 nonadjacent are 132 and 231, therefore a(2)=2. - Jon Perry, Jun 08 2003
Number of multiplications performed when computing the determinant of an n X n matrix by definition. - Mats Granvik, Sep 12 2008
Sum of the length of all cycles (excluding fixed points) in all permutations of [n]. - Olivier Gérard, Oct 23 2012
Number of permutations of n distinct objects (ABC...) 1 (one) times >>("-", A, AB, ABC, ABCD, ABCDE, ..., ABCDEFGHIJK, infinity) and one after the other to resemble motif: A (1) AB (1-1), AAB (2-1), AAAB (3-1), AAAAB (4-1), AAAAAB (5-1), AAAAAAB (6-1), AAAAAAAB (7-1), AAAAAAAAB (8-1) etc.,>> "1(one) fixed point". Example:motif: AAAB (or BBBA) 12 * one (1) fixed point etc. Let: AAAB ................ 'A'BCD 1. 'A'BDC 2. 'A'CBD 3. ACDB 'A'DBC 4. 'A'DCB B'A'CD 5. B'A'DC 6. BCAD 7. BCDA BD'A'C 8. BDCA C'A'BD 9. C'A'DB CB'A'D 10. CBDA CDAB CDBA D'A'BC 11. DACB DB'A'C 12. DBCA DCAB DCBA. - Zerinvary Lajos, Nov 27 2009 (does anybody understand what this is supposed to say? - Joerg Arndt, Jan 10 2015)
a(n) is the number of ways to arrange n books on two bookshelves so that each shelf receives at least one book. - Geoffrey Critzer, Feb 21 2010
a(n) = number whose factorial base representation (A007623) begins with digit {n-1} and is followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 10, 200, 3000, 40000, 500000, 6000000, 70000000, 800000000, 9000000000, A0000000000, B00000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

Links

Crossrefs

Column 2 of A257503 (apart from initial zero. Equally, row 2 of A257505).
Cf. A001286 (same sequence divided by 2).
Cf. A000142, A007623, A018931, A052849.
Cf. A001563. - Zerinvary Lajos, Aug 27 2008
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

Programs

Formula

a(n) = n! * (n-1).
E.g.f.: x^2/(1-x)^2. - Geoffrey Critzer, Feb 21 2010
a(n) = 2 * A001286(n).
a(n) = A001563(n) - A000142(n). - Antti Karttunen, May 07 2015, hinted by crossref left by Lajos.
From Amiram Eldar, Jul 11 2020: (Start)
Sum_{n>=2} 1/a(n) = Ei(1) + 2 - e - gamma = A091725 + 2 - A001113 - A001620.
Sum_{n>=2} (-1)^n/a(n) = gamma - Ei(-1) - 1/e = A001620 + A099285 - A068985. (End)

Extensions

Last term a(19) corrected by Harry J. Smith, Aug 02 2009

A052571 E.g.f. x^3/(1-x)^2.

Original entry on oeis.org

0, 0, 0, 6, 48, 360, 2880, 25200, 241920, 2540160, 29030400, 359251200, 4790016000, 68497228800, 1046139494400, 16999766784000, 292919058432000, 5335311421440000, 102437979291648000, 2067966706950144000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

For n >= 3, a(n) = number whose factorial base representation (A007623) begins with digit {n-2} followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 0, 0, 100, 2000, 30000, 400000, 5000000, 60000000, 700000000, 8000000000, 90000000000, A00000000000, B000000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

Links

Crossrefs

Column 5 of A257503 (apart from zero terms. Equally, row 5 of A257505).
Cf. A000142, A007623, A005990, A090672.
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A091725, A099285.

Programs

  • Magma
    [0,0] cat [n*(n+1)*(n+2)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011
  • Maple
    spec := [S,{S=Prod(Z,Z,Z,Sequence(Z),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
[seq (n*(n+1)*(n+2)*n!,n=0..17)]; # Zerinvary Lajos, Nov 25 2006
a:=n->add((n!),j=1..n-2):seq(a(n), n=0..21); # Zerinvary Lajos, Aug 27 2008
G(x):=x^3/(1-x)^2: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # Zerinvary Lajos, Apr 01 2009
  • Mathematica
    Table[Sum[n!, {i, 3, n}], {n, 0, 19}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[x^3/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 27 2025 *)
  • Scheme
    (define (A052571 n) (if (< n 2) 0 (* (- n 2) (A000142 n)))) ;; Antti Karttunen, May 07 2015

Formula

E.g.f.: x^3/(-1+x)^2.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(3)=6, (1-n^2)*a(n)+(-2+n)*a(n+1)=0}.
For n >= 2, a(n) = (n-2)*n!.
a(n+2) = n*(n+1)*(n+2)*n!. - Zerinvary Lajos, Nov 25 2006
a(n) = 3*A090672(n-2) = 6*A005990(n-2). - Zerinvary Lajos, May 11 2007
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=3} 1/a(n) = 9/4 - e - gamma/2 + Ei(1)/2 = 9/4 - A001113 - (1/2)*A001620 + (1/2)*A091725.
Sum_{n>=3} (-1)^(n+1)/a(n) = -1/4 + gamma/2 - Ei(-1)/2 = -1/4 + (1/2)*A001620 + (1/2)*A099285. (End)

A052572 Expansion of e.g.f.: (1+2*x-2*x^2)/(1-x)^2.

Original entry on oeis.org

1, 4, 10, 36, 168, 960, 6480, 50400, 443520, 4354560, 47174400, 558835200, 7185024000, 99632332800, 1482030950400, 23538138624000, 397533007872000, 7113748561920000, 134449847820288000, 2676192208994304000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

a(n) equals the permanent of the (n+1) X (n+1) matrix whose entry directly below the entry in the top right corner is 3, and all of whose other entries are 1. - John M. Campbell, May 25 2011
In factorial base representation (A007623) the terms are written as: 1, 20, 120, 1200, 12000, 120000, ... From a(2) = 10 = "120" onward each term begins always with "120", followed by n-2 additional zeros. - Antti Karttunen, Sep 24 2016

Links

Crossrefs

Essentially twice A038720.
Row 7 of A276955, from a(2)=10 onward.
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A000142.

Programs

  • Magma
    A052572:= func< n | n eq 0 select 1 else (n+3)*Factorial(n) >; // G. C. Greubel, May 11 2025
  • Maple
    spec := [S,{S=Prod(Union(Z,Z,Sequence(Z)),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    With[{nn=20},CoefficientList[Series[(1+2x-2x^2)/(1-x)^2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 03 2020 *)
Table[If[n==0, 1, (n+3)*n!], {n,0,30}] (* G. C. Greubel, May 11 2025 *)
  • SageMath
    def A052572(n): return 1 if n==0 else (n+3)*factorial(n) # G. C. Greubel, May 11 2025

Formula

E.g.f.: (1 + 2*x - 2*x^2)/(1 - x)^2.
Recurrence: (n+2)*a(n) = n*(n+3)*a(n-1), with a(0) = 1, a(1) = 4, a(2) = 10.
a(n) = (n+3)*n! for n > 0.
For n <= 1, a(n) = (n+1)^2, for n > 1, a(n) = (n+1)! + 2*n! - Antti Karttunen, Sep 24 2016
From Amiram Eldar, Nov 06 2020: (Start)
Sum_{n>=0} 1/a(n) = e - 4/3.
Sum_{n>=0} (-1)^n/a(n) = 8/3 - 5/e. (End)

A052520 Number of pairs of sequences of cardinality at least 2.

Original entry on oeis.org

0, 0, 0, 0, 24, 240, 2160, 20160, 201600, 2177280, 25401600, 319334400, 4311014400, 62270208000, 958961203200, 15692092416000, 271996268544000, 4979623993344000, 96035605585920000, 1946321606541312000, 41359334139002880000, 919636959090769920000, 21356013827774545920000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

Programs

  • GAP
    Concatenation([0,0,0,0], List([4..20], n-> (n-3)*Factorial(n))); # G. C. Greubel, May 13 2019
  • Magma
    [n le 3 select 0 else (n-3)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 13 2019
  • Maple
    spec := [S,{B=Sequence(Z,2 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Table[Sum[n!, {i,4,n}], {n, 0, 19}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[x^4/(x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 03 2016 *)
  • PARI
    {a(n) = if(n<4, 0, (n-3)*n!)}; \\ G. C. Greubel, May 13 2019
  • Sage
    [0,0,0,0]+[(n-3)*factorial(n) for n in (4..20)] # G. C. Greubel, May 13 2019

Formula

E.g.f.: x^4/(1-x)^2.
(n-3)*a(n+1) + (2+n-n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = 0, a(4) = 24.
a(n) = (n-3)*n!, n>2.
a(n) = (n+1)!*(n-3)/(n+1), n>2. - Gary Detlefs, Oct 02 2011
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=4} 1/a(n) = 59/36 - 2*e/3 - gamma/6 + Ei(1)/6 = 59/36 - (2/3)*A001113 - (1/6)*A001620 + A091725/2.
Sum_{n>=4} (-1)^n/a(n) = 1/36 - 1/(3*e) + gamma/6 - Ei(-1)/6 = 1/36 - (1/3)*A068985 + (1/6)*A001620 + (1/6)*A099285. (End)

A052521 Number of pairs of sequences of cardinality at least 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 10080, 120960, 1451520, 18144000, 239500800, 3353011200, 49816166400, 784604620800, 13076743680000, 230150688768000, 4268249137152000, 83230858174464000, 1703031405723648000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

Programs

  • GAP
    Concatenation([0,0,0,0,0,0], List([6..20], n-> (n-5)*Factorial(n))); # G. C. Greubel, May 13 2019
  • Magma
    [n le 5 select 0 else (n-5)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 13 2019
  • Maple
    spec := [S,{B=Sequence(Z,3 <= card), S=Prod(B,B)},labeled]: # Pairs spec
seq(combstruct[count](spec, size=n), n=0..20);
  • Mathematica
    Table[If[n<6, 0, (n-5)*n!], {n,0,20}] (* G. C. Greubel, May 13 2019 *)
  • PARI
    {a(n) = if(n<6, 0, (n-5)*n!)}; \\ G. C. Greubel, May 13 2019
  • Sage
    [0,0,0,0,0,0]+[(n-5)*factorial(n) for n in (6..20)] # G. C. Greubel, May 13 2019

Formula

E.g.f.: x^6/(1-x)^2.
(n-5)*a(n+1) + (4 + 3*n - n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 720.
a(n) = (n-5)*n!.
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=6} 1/a(n) = 5477/7200 - 17*e/60 - gamma/120 + Ei(1)/120 = 5477/7200 - (17/60)*A001113 - (1/120)*A001620 + A091725/120.
Sum_{n>=6} (-1)^n/a(n) = 403/7200 - 1/(6*e) + gamma/120 - Ei(-1)/120 = 403/7200 - (1/6)*A068985 + (1/120)*A001620 + (1/120)*A099285. (End)

A282822 a(n) = (n - 4)*n! for n>=0.

Original entry on oeis.org

-4, -3, -4, -6, 0, 120, 1440, 15120, 161280, 1814400, 21772800, 279417600, 3832012800, 56043187200, 871782912000, 14384418048000, 251073478656000, 4623936565248000, 89633231880192000, 1824676506132480000, 38926432130826240000, 868546016919060480000
Offset: 0

Views

Author

Bruno Berselli, Feb 22 2017

Keywords

Crossrefs

Cf. A034865.
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

Programs

  • Mathematica
    Table[(n - 4) n!, {n, 0, 30}] (* or *)
RecurrenceTable[{a[0] == -4, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}]

Formula

E.g.f.: -(4 - 5*x)/(1 - x)^2.
a(n) = n*a(n-1) + n!, with n>0, a(0)=-4.
a(n) = 2*A034865(n) for n>3.
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=5} 1/a(n) = 313/288 - 5*e/12 - gamma/24 + Ei(1)/24 = 313/288 - (5/12)*A001113 - (1/24)*A001620 + A091725/24.
Sum_{n>=5} (-1)^(n+1)/a(n) = -25/288 + 1/(6*e) + gamma/24 - Ei(-1)/24 = -25/288 - (1/6)*A068985 + (1/24)*A001620 + (1/24)*A099285. (End)

A052644 Expansion of e.g.f. (1+3x-3x^2)/(1-x)^2.

Original entry on oeis.org

1, 5, 12, 42, 192, 1080, 7200, 55440, 483840, 4717440, 50803200, 598752000, 7664025600, 105859353600, 1569209241600, 24845812992000, 418455797760000, 7469435990016000, 140852221526016000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. sequences with formula (n + k)*n! listed in A282466.

Programs

  • Maple
    spec := [S,{S=Prod(Sequence(Z),Union(Z,Z,Z,Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    With[{nn=20},CoefficientList[Series[(1+3x-3x^2)/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 06 2014 *)

Formula

E.g.f.: -(-3*x+3*x^2-1)/(-1+x)^2
Recurrence: {a(0)=1, a(1)=5, (-n^2-6*n-5)*a(n)+(n+4)*a(n+1)=0, a(2)=12}.
a(n) = (n+4)*n!, n>0.
G.f.: G(0) where G(k) = 1 + x*(k+1)*(k+4)/(1 - 1/(1 + (k+4)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012
From Amiram Eldar, Nov 06 2020: (Start)
Sum_{n>=0} 1/a(n) = 27/4- 2*e.
Sum_{n>=0} (-1)^n/a(n) = 27/4 - 16/e. (End)
Showing 1-9 of 9 results.

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