A094258 -id:A094258 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

			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 + ...

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.

T. D. Noe, Table of n, a(n) for n = 0..100
J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]
Loïc Foissy, The antipode of of [sic] a Com-PreLie Hopf algebra, arXiv:2406.01120 [math.CO], 2024. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 30.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
L. B. W. Jolley, Summation of Series, Dover, 1961.
I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446v2 [math.CO], 18 May 2008.
C. Lanczos, Applied Analysis (Annotated scans of selected pages).
Rezsö L. Lovas and István Mezö, On an exotic topology of the integers, arXiv:1008.0713 [math.GN], 2010. See p. 4.
Daniel J. Mundfrom, A problem in permutations: the game of `Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages).
A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
Dennis P. Walsh, The number of permutations with no k-cycles.
Eric Weisstein's World of Mathematics, Distinguishing Number.
Eric Weisstein's World of Mathematics, Exponential Integral.

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.

List([0..20], n-> n*Factorial(n) ); # G. C. Greubel, Dec 30 2019

a001563 n = a001563_list !! n
a001563_list = zipWith (-) (tail a000142_list) a000142_list
-- Reinhard Zumkeller, Aug 05 2013

[Factorial(n+1)-Factorial(n): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
A001563 := n->n*n!;
```

Table[n!n,{n,0,25}] (* Harvey P. Dale, Oct 03 2011 *)

{a(n) = if( n<0, 0, n * n!)} /* Michael Somos, Dec 11 2002 */

[n*factorial(n) for n in (0..20)] # G. C. Greubel, Dec 30 2019

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

A094304 Sum of all possible sums formed from all but one of the previous terms, starting 1.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Apr 29 2004

nonn

Apart from initial 1, same sequence as A001563. Additive analog of A057438.

a(1) = 1, for n >= 2: a(n) = sum of previous terms * (n-2) = (Sum_(i=1...n-2) a(i)) * (n-2). a(n) = A001563(n-2) = A094258(n-1) for n >= 3. - Jaroslav Krizek, Oct 16 2009

			a(2) = 0 as there is only one previous term and empty sum is taken to be 0.
a(4) = (a(1) +a(2))+ (a(1) +a(3)) + (a(2) +a(3)) = (1+0) +(1+1) +(0+1) = 4.
a(5) = (a(1)+a(2)+a(3)) +(a(1)+a(2)+ a(4)) +(a(1)+a(3)+a(4)) +(a(2)+a(3)+a(4)) = (1+0+1) +(1+0+4) +(1+1+4) +(0+1+4) = 2 + 5 + 6 + 5 = 18.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A001563, A057438, A122974.

a := n -> (n-2)*(n-2)!: 1,seq(a(n), n=2..23); # Emeric Deutsch, May 01 2008

In[2]:= l = {1}; Do[k = Length[l] - 1; p = Plus @@ Flatten[Select[Subsets[l], Length[ # ]==k& ]]; AppendTo[l, p], {n, 20}]; l (* Ryan Propper, May 28 2006 *)

v=vector(30);v[1]=1;v[2]=0;for(n=3,#v,s=0;for(i=1,2^(n-1)-1, vb=binary(i); if(hammingweight(vb)==n-2,s=s+sum(j=1,#vb, if(vb[j], v[n-#vb+j-1]))));v[n]=s;print1(s,",")) /* Ralf Stephan, Sep 22 2013 */

a(n) = (n-2)!(n-2) for n>=2. - Emeric Deutsch, May 01 2008
G.f.: x*T(0), 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, Nov 10 2013
a(n) = S1(n,1) - S1(n-1,1), where S1 are the unsigned Stirling cycle numbers. - Peter Luschny, Apr 10 2016
a(n) = A122974(n-1,n-1). - Alois P. Heinz, Nov 24 2019

Edited by N. J. A. Sloane, May 29 2006

A188914 a(n) = n*n! + 1 = (n+1)! - n! + 1.

Original entry on oeis.org

1, 2, 5, 19, 97, 601, 4321, 35281, 322561, 3265921, 36288001, 439084801, 5748019201, 80951270401, 1220496076801, 19615115520001, 334764638208001, 6046686277632001, 115242726703104001, 2311256907767808001, 48658040163532800001, 1072909785605898240001

Offset: 0

John M. Campbell, Apr 17 2011

It is unknown if all numbers of the form n*n!+1 are squarefree. n*n!+1 is squarefree for 0 < n < 52. It is unknown if there exist infinitely many primes of the form n*n!+1. For primes in this sequence, see A049984.

Cf. A001563, A049984, A094258.

Mathematica
```
Table[(n*Factorial[n])+1,{n,0,30}]
```

a(n) = n*n! + 1; \\ Michel Marcus, Aug 03 2022

E.g.f.: exp(x) + x/(1 - x)^2. - Stefano Spezia, Aug 03 2022

a(0)=1 prepended by Alois P. Heinz, Aug 03 2022

A230339 Numerator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Original entry on oeis.org

0, 1, 1, 19, 17, 55, 83, 119, 82, 73, 95, 121, 227, 559, 679, 815, 484, 1139, 443, 171, 295, 2023, 2299, 2599, 1462, 3275, 3653, 451, 749, 551, 5455, 5983, 3272, 7139, 7769, 8435, 1523, 3293, 3553, 11479, 6170, 13243, 14189, 15179, 8107, 5765

Offset: 0

Jean-François Alcover, Oct 16 2013

			1/(1*2*3*4) + 1/(2*3*4*5) + 1/(3*4*5*6) = 19/360, so a(3) = 19.
The rationals r(n) = a(n)/A230340(n) begin: 0, 1/24, 1/20, 19/360, 17/315, 55/1008, 83/1512, 119/2160, 82/1485, 73/1320, 95/1716, 121/2184, 227/4095, 559/10080, 679/12240, 815/14688, ... - _Wolfdieter Lang_, Mar 08 2018

L. B. W. Jolley, Summation of Series, Second revised ed., Dover, 1961, p.38, (202) and (201).

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Pochhammer Symbol

Cf. A001563, A052762, A094258, A125650, A230328, A230340 (denominators).

a[n_] := Numerator[1/18 - 1/(3*(n+1)*(n+2)*(n+3))]; Table[a[n], {n, 0, 100}]

a(n) = numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) \\ Colin Barker, Jul 30 2019

Numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) (from the generic formula Sum_{k=1..n} 1/Pochhammer(k, m) = 1/((m-1)*(m-1)!) - 1/((m-1)*Pochhammer(n+1, m-1)) with m = 4).
G.f. for the rationals r(n) = (1/18)*n*(11+n^2+6*n)/((1+n)*(n+2)*(n+3)) = a(n)/A230340(n): (1/18)*(1 - hypergeometric([1, 3], [4], -x/(1-x)))/(1-x) = (6*x - 15*x^2 + 11*x^3 +  6*(1 - 3*x + 3*x^2 - x^3)*log(1-x))/(36*x^3*(1-x)). - Wolfdieter Lang, Mar 08 2018
a(n) = numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))). - Colin Barker, Jul 30 2019

A230340 Denominator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Original entry on oeis.org

1, 24, 20, 360, 315, 1008, 1512, 2160, 1485, 1320, 1716, 2184, 4095, 10080, 12240, 14688, 8721, 20520, 7980, 3080, 5313, 36432, 41400, 46800, 26325, 58968, 65772, 8120, 13485, 9920, 98208, 107712, 58905, 128520, 139860, 151848, 27417, 59280

Offset: 0

Jean-François Alcover, Oct 16 2013

			1/(1*2*3*4) + 1/(2*3*4*5) + 1/(3*4*5*6) = 19/360, so a(3) = 360.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Pochhammer Symbol

Cf. A001563, A052762, A094258, A125650, A230328, A230339 (numerators).

[Denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))):n in [0..100]]; // Marius A. Burtea, Jul 30 2019

a[n_] := Denominator[1/18 - 1/(3*(n+1)*(n+2)*(n+3))]; Table[a[n], {n, 0, 100}]

a(n) = denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) \\ Colin Barker, Jul 30 2019

Denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))).

A280318 a(n) is the n-th permutation generated by Heap's algorithm, represented by row number of A055089.

Original entry on oeis.org

0, 1, 3, 2, 4, 5, 11, 10, 8, 9, 7, 6, 12, 13, 15, 14, 16, 17, 23, 22, 20, 21, 19, 18, 93, 92, 94, 95, 90, 91, 78, 79, 81, 80, 82, 83, 89, 88, 86, 87, 85, 84, 74, 75, 73, 72, 77, 76, 52, 53, 48, 49, 51, 50, 71, 70, 68, 69, 67, 66, 55, 54, 59, 58

Offset: 0

Tilman Piesk, Dec 31 2016

This is a permutation of the nonnegative integers. It divides naturally in sections of factorial length, so it can be seen as a triangle with row lengths A094258:
0,
1,
3, 2, 4, 5,
11, 10, 8, 9, 7, 6, 12, 13, 15, 14, 16, 17, 23, 22, 20, 21, 19, 18...
Compare A280319 for Steinhaus-Johnson-Trotter algorithm, which is a triangle of finite permutations rather than one infinite permutation.

			Example for the first 24 entries of the sequence. On the right are the permutations of {1,2,3,4} in the order generated by the Heap's algorithm:
   n    rev colex        a(n)   Heap's
   0     1 2 3 4          0     1 2 3 4
   1     2 1 3 4          1     2 1 3 4
   2     1 3 2 4          3     3 1 2 4
   3     3 1 2 4          2     1 3 2 4
   4     2 3 1 4          4     2 3 1 4
   5     3 2 1 4          5     3 2 1 4
   6     1 2 4 3         11     4 2 1 3
   7     2 1 4 3         10     2 4 1 3
   8     1 4 2 3          8     1 4 2 3
   9     4 1 2 3          9     4 1 2 3
  10     2 4 1 3          7     2 1 4 3
  11     4 2 1 3          6     1 2 4 3
  12     1 3 4 2         12     1 3 4 2
  13     3 1 4 2         13     3 1 4 2
  14     1 4 3 2         15     4 1 3 2
  15     4 1 3 2         14     1 4 3 2
  16     3 4 1 2         16     3 4 1 2
  17     4 3 1 2         17     4 3 1 2
  18     2 3 4 1         23     4 3 2 1
  19     3 2 4 1         22     3 4 2 1
  20     2 4 3 1         20     2 4 3 1
  21     4 2 3 1         21     4 2 3 1
  22     3 4 2 1         19     3 2 4 1
  23     4 3 2 1         18     2 3 4 1

Tilman Piesk, Table of n, a(n) for n = 0..5039
Tilman Piesk, Calculation in Python
Wikipedia, Heap's algorithm

Cf. A055089, A094258, A280319.

A109075 Number of primes which use each of 0-to-n decimal digits exactly once.

Original entry on oeis.org

0, 0, 0, 0, 16, 0, 0, 2668, 0, 0

Offset: 0

Zak Seidov, Jun 21 2005

There are exactly 16 five-digit primes using decimal digits 0-to-4 exactly once: A109176 and 2668 eight-digit primes using each of 0-to-7 decimal digits exactly once: A109177, A109178. Cf. A094258.

Cf. A094258, A109176, A109177, A109178.

A177262 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} starting with exactly k consecutive integers (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 18, 4, 1, 1, 96, 18, 4, 1, 1, 600, 96, 18, 4, 1, 1, 4320, 600, 96, 18, 4, 1, 1, 35280, 4320, 600, 96, 18, 4, 1, 1, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1, 3265920, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1, 36288000, 3265920, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1

Offset: 1

Emeric Deutsch, May 15 2010

			T(4,2)=4 because we have 1243, 2314, 3412, and 3421.
Triangle starts:
      1;
      1,    1;
      4,    1,   1;
     18,    4,   1,  1;
     96,   18,   4,  1,  1;
    600,   96,  18,  4,  1,  1;
   4320,  600,  96, 18,  4,  1,  1;
  35280, 4320, 600, 96, 18,  4,  1,  1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000142 (row sums), A005165, A007489, A094258.

A177262:= func< n,k | k eq n select 1 else (n-k)*Factorial(n-k) >;
[A177262(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, May 18 2024

T := proc (n, k) if k = n then 1 elif k < n then factorial(n-k)*(n-k) else 0 end if end proc: for n to 11 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

T[n_, k_]:= (n-k+1)! -(n-k)! +Boole[k==n];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 18 2024 *)

def A177262(n,k): return (n-k)*factorial(n-k) + int(k==n)
flatten([[A177262(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, May 18 2024

T(n, k) = (n-k)*(n-k)! if k < n, otherwise T(n,n) = 1.
T(n, 1) = A094258(n) = (n-1)!(n-1).
Sum_{k=1..n} T(n, k) = A000142(n) (row sums).
Sum_{k=1..n} k*T(n,k) = Sum_{j=1..n} j! = A007489(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A005165(n). - G. C. Greubel, May 18 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

A094304 Sum of all possible sums formed from all but one of the previous terms, starting 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A188914 a(n) = n*n! + 1 = (n+1)! - n! + 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A230339 Numerator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A230340 Denominator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A280318 a(n) is the n-th permutation generated by Heap's algorithm, represented by row number of A055089.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A109075 Number of primes which use each of 0-to-n decimal digits exactly once.

Views

Author

Keywords