A245334 -id:A245334 - cp's OEIS frontend

A049398 a(n) = (n+9)!/9!.

1, 10, 110, 1320, 17160, 240240, 3603600, 57657600, 980179200, 17643225600, 335221286400, 6704425728000, 140792940288000, 3097444686336000, 71241227785728000, 1709789466857472000, 42744736671436800000, 1111363153457356800000, 30006805143348633600000

Offset: 0

Wolfdieter Lang

The p=9 member of the p-family of sequences {(n+p-1)!/p!}, n >= 1.

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=10) ~ exp(-x)/x*(1 - 10/x + 110/x^2 - 1320/x^3 + 17160/x^4 - 240240/x^5 + 3603600/x^6 - ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A049388, A049389.

Cf. A130534, A163931, A173333, A245334.

a049398 = (flip div 362880) . a000142 . (+ 9)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+9)/362880: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_] := (n + 9)!/9!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)

PARI
```
a(n) = (n+9)!/9!
```

E.g.f.: 1/(1-x)^10.
a(n) = A173333(n+9,9). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+9,n) / 10. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 362880*e - 986409.
Sum_{n>=0} (-1)^n/a(n) = 133497 - 362880/e. (End)

A027810 a(n) = (n+1)*binomial(n+5, 5).

Original entry on oeis.org

1, 12, 63, 224, 630, 1512, 3234, 6336, 11583, 20020, 33033, 52416, 80444, 119952, 174420, 248064, 345933, 474012, 639331, 850080, 1115730, 1447160, 1856790, 2358720, 2968875, 3705156, 4587597, 5638528, 6882744, 8347680, 10063592, 12063744, 14384601, 17066028

Offset: 0

Thi Ngoc Dinh (via R. K. Guy)

Number of 11-subsequences of [ 1, n ] with just 5 contiguous pairs.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, Carus Mathematical Monographs No. 14, John Wiley and Sons, 1963, pp. 1-8.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Partial sums of A051843.
Cf. A093563 ((6, 1) Pascal, column m=6).
Cf. A007318, A000142, A245334.

a027810 n = (n + 1) * a007318' (n + 5) 5
-- Reinhard Zumkeller, Aug 31 2014

[(n+1)*Binomial(n+5,5): n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

[n*(n-1)*(n-2)*(n-3)*(n-4)^2/120: n in [5..40]]; // Vincenzo Librandi, Jul 30 2014

[seq(n*(n-1)*(n-2)*(n-3)*(n-4)^2/5!,n=5..33)]; # Zerinvary Lajos, Oct 19 2006

Table[(n+1)Binomial[n+5,5],{n,0,30}] (* Harvey P. Dale, Jul 29 2014 *)
CoefficientList[Series[(1 + 5 x)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

a(n)=n*(n^5+16*n^4+100*n^3+310*n^2+499*n+394)/120+1 \\ Charles R Greathouse IV, Sep 28 2015

G.f.: (1+5*x)/(1-x)^7.
a(n) = A245334(n+5, 5)/A000142(5). - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 5*Pi^2/6 - 1025/144.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^2/12 - 160*log(2)/3 + 4865/144. (End)

Two redundant formulas deleted by N. J. A. Sloane, Jul 30 2014

A051431 a(n) = (n+10)!/10!.

Original entry on oeis.org

1, 11, 132, 1716, 24024, 360360, 5765760, 98017920, 1764322560, 33522128640, 670442572800, 14079294028800, 309744468633600, 7124122778572800, 170978946685747200, 4274473667143680000, 111136315345735680000, 3000680514334863360000, 84019054401376174080000

Offset: 0

Wolfdieter Lang

The p=10 member of the p-family of sequences {(n+p-1)!/p!}, n >= 1.

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=11) ~ exp(-x)/x*(1 - 11/x + 132/x^2 - 1716/x^3 + 24024/x^4 - 360360/x^5 + 5765760/x^6 - ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398.

Cf. A130534, A163931, A173333, A245334.

a051431 = (flip div 3628800) . a000142 . (+ 10)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+10)/3628800: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_] := (n + 10)!/10!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)

a(n) = (n+10)!/10!
E.g.f.: 1/(1-x)^11.
a(n) = A173333(n+10,10). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+10,n) / 11. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 3628800*e - 9864100.
Sum_{n>=0} (-1)^n/a(n) = 3628800/e - 1334960. (End)

A070960 a(1) = 1; a(n) = n!*(3/2) for n>=2.

Original entry on oeis.org

1, 3, 9, 36, 180, 1080, 7560, 60480, 544320, 5443200, 59875200, 718502400, 9340531200, 130767436800, 1961511552000, 31384184832000, 533531142144000, 9603560558592000, 182467650613248000, 3649353012264960000, 76636413257564160000, 1686001091666411520000, 38778025108327464960000

Offset: 1

Koksal Karakus (karakusk(AT)hotmail.com), May 24 2002

Let g be a permutation of [1..n] having, say, j_i cycles of length i, with Sum_i i*j_i = n; sequence gives Sum_{g} Sum_{i} (j_1 + j_2). - N. J. A. Sloane, Jul 22 2009

a(n) is the greatest integer that can be obtained from the integers {1, 2, 3, ..., n} using each number at most once and the operators +, -, *, /.

			a(5) = 180 because the greatest number we can obtain using 1, 2, 3, 4, 5 is 180 which is (1+2)*3*4*5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..400
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for similar sequences.

Cf. A000142, A001710, A060315.

Cf. A245334.

a070960 n = if n == 1 then 1 else 3 * a000142 n `div` 2
a070960_list = map (flip div 2) fs where fs = 3 : zipWith (*) [2..] fs
-- Reinhard Zumkeller, Aug 31 2014

s=3;lst={1, s};Do[s+=n*s+s;AppendTo[lst, s], {n, 1, 5!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Join[{1},(3*Range[2,20]!)/2] (* Harvey P. Dale, Jun 15 2022 *)

a(n) = if (n==1, 1, n!*3/2); \\ Michel Marcus, Dec 03 2022

E.g.f.: x*(2+x)/(1-x)/2. - Vladeta Jovovic, Dec 15 2002
a(n) = A245334(n,n-2), n > 1. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=1} 1/a(n) = (2*e-1)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - 2/(3*e). (End)
a(n) = A000142(n) + A001710(n) for n>=2. - Alois P. Heinz, Feb 20 2024

Edited by N. J. A. Sloane, Jul 22 2009

A027818 a(n) = (n+1)*binomial(n+6,6).

Original entry on oeis.org

1, 14, 84, 336, 1050, 2772, 6468, 13728, 27027, 50050, 88088, 148512, 241332, 379848, 581400, 868224, 1268421, 1817046, 2557324, 3542000, 4834830, 6512220, 8665020, 11400480, 14844375, 19143306, 24467184, 31011904, 39002216, 48694800, 60381552, 74393088, 91102473

Offset: 0

Thi Ngoc Dinh (via R. K. Guy)

Number of 13-subsequences of [ 1, n ] with just 6 contiguous pairs.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A093564 ((7, 1) Pascal, column m=7). Partial sums of A050403.

Cf. A062190, A007318, A000142, A245334.

List([0..30], n-> (n+1)*Binomial(n+6,6)); # G. C. Greubel, Aug 29 2019

a027818 n = (n + 1) * a007318' (n + 6) 6
-- Reinhard Zumkeller, Aug 31 2014

[(n+1)*Binomial(n+6,6): n in [0..30]]; // G. C. Greubel, Aug 29 2019

seq((n+1)*binomial(n+6,6), n=0..30); # Zerinvary Lajos, Oct 19 2006

Table[(n+1)*Binomial[n+6,6], {n,0,30}] (* G. C. Greubel, Aug 29 2019 *)

a(n) = (n+1)*binomial(n+6,6) \\ Charles R Greathouse IV, Jun 11 2015

[(n+1)*binomial(n+6,6) for n in (0..30)] # G. C. Greubel, Aug 29 2019

G.f.: (1+6*x)/(1-x)^8.
a(n) = A245334(n+6,6)/A000142(6). - Reinhard Zumkeller, Aug 31 2014
E.g.f.: (7! +9360*x +20520*x^2 +15000*x^3 +4650*x^4 +666*x^5 +43*x^6 + x^7)*exp(x)/7!. - G. C. Greubel, Aug 29 2019
From Amiram Eldar, Jan 28 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi^2 - 5269/600.
Sum_{n>=0} (-1)^n/a(n) = Pi^2/2 - 512*log(2)/5 + 40189/600. (End)

A047927 a(n) = n(n-1)(n-2)^2.

Original entry on oeis.org

0, 6, 48, 180, 480, 1050, 2016, 3528, 5760, 8910, 13200, 18876, 26208, 35490, 47040, 61200, 78336, 98838, 123120, 151620, 184800, 223146, 267168, 317400, 374400, 438750, 511056, 591948, 682080, 782130, 892800, 1014816, 1148928, 1295910, 1456560, 1631700, 1822176

Offset: 2

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002417, A059238, A245334.

a047927 n = if n == 2 then 0 else a245334 n 3
-- Reinhard Zumkeller, Aug 31 2014

[n*(n-1)*(n-2)^2: n in [2..40]]; // Vincenzo Librandi, May 02 2011

a[n_] := n*(n-1)*(n-2)^2; Array[a, 50, 2] (* Amiram Eldar, Jan 15 2023 *)

a(n)=n*(n-1)*(n-2)^2 \\ Charles R Greathouse IV, Oct 07 2015

From R. J. Mathar, May 01 2014: (Start)
G.f.: -6*x^3*(1+3*x) / (x-1)^5.
a(n) = 6*A002417(n-2). (End)
a(n) = A245334(n,3), n > 2. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=3} 1/a(n) = Pi^2/12 - 5/8.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 - 2*log(2) + 9/8. (End)

Offset changed from 0 to 2 by Vincenzo Librandi, May 02 2011

A056003 a(n) = (n+1)*binomial(n+8, 8).

Original entry on oeis.org

1, 18, 135, 660, 2475, 7722, 21021, 51480, 115830, 243100, 481338, 906984, 1637610, 2848860, 4796550, 7845024, 12503007, 19468350, 29683225, 44401500, 65270205, 94427190, 134617275, 189329400, 262957500, 360988056, 490217508, 659002960, 877549860, 1158240600

Offset: 0

Barry E. Williams, Jun 12 2000

Original name: A second-order recursive sequence.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Partial sums of A056117.
Cf. A093644 ((9, 1) Pascal, column m=9).
Cf. A000142, A007318, A052206, A245334, A254142 (partial sums).

a056003 n = (n + 1) * a007318' (n + 8) 8
-- Reinhard Zumkeller, Aug 31 2014

a:=n->(sum((numbcomp(n,9)), j=9..n)):seq(a(n), n=9..35); # Zerinvary Lajos, Aug 26 2008

a[n_] := (n+1)*Binomial[n+8, 8]; Array[a, 50, 0] (* Amiram Eldar, Jan 15 2023 *)

a(n) = (n+1)*binomial(n+8, 8) \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+8*x)/(1-x)^10.
a(n) = A245334(n+8,8)/A000142(8). - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 4*Pi^2/3 - 266681/22050.
Sum_{n>=0} (-1)^n/a(n) = 2*Pi^2/3 - 38656*log(2)/105 + 611409/2450. (End)

A111063 a(0) = 1; a(n) = (n-1)*a(n-1) + n.

Original entry on oeis.org

1, 1, 3, 9, 31, 129, 651, 3913, 27399, 219201, 1972819, 19728201, 217010223, 2604122689, 33853594971, 473950329609, 7109254944151, 113748079106433, 1933717344809379, 34806912206568841, 661331331924807999, 13226626638496160001, 277759159408419360043

Offset: 0

N. J. A. Sloane, Oct 08 2005

From Frank Ruskey, Nov 24 2009: (Start)
If the initial 1 were deleted, the recurrence relation becomes a(n) = n+1+n*a(n-1) with a(0) = 1. Furthermore, it can then be shown that a(n) is the number of nonempty subsets of binary strings with n 1's and 2 0's that are closed under the operation of replacing the leftmost 01 with 10. Taking the maximal elements under this relation,
a(2) = 9 = |{0011},{0101},{1001},{1010},{1100},{0110}, {0110,1001},{0101,0110},{0011,0110}|.
We also have the e.g.f. (1+x)/(1-x) e^x and the formula a(n) = -1 + 2*n!*sum_{j=0..n} 1/j!. (End)
a(n+1) = sum of n-th row in triangle A245334. - Reinhard Zumkeller, Aug 30 2014 [A-number corrected by N. J. A. Sloane, May 03 2017]
Eigensequence of triangle with (1, 2, 3, ...) as the right and left borders and the rest zeros. - Gary W. Adamson, Aug 01 2016
The following remarks apply to the sequence without the initial term a(0) = 1: For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k. It follows that for each k the sequence a(n) taken modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes 1, 3, 9, 1, 9, 1, 3, 9, 1, 9, ... with period 5. Cf. A000522. - Peter Bala, Nov 19 2017

F. Drewes et al., Tight Bounds for Cut-Operations on Deterministic Finite Automata, in Lecture Notes in Computer Science, Volume 9288 2015, Machines, Computations, and Universality, 7th International Conference, MCU 2015, Famagusta, North Cyprus, September 9-11, 2015, Editors: Jerome Durand-Lose, Benedek Nagy, ISBN: 978-3-319-23110-5 (Print) 978-3-319-23111-2 (Online). ["In the On-Line Encyclopedia of Integer Sequences (OEIS) this matches the sequence A111063."]

Reinhard Zumkeller, Table of n, a(n) for n = 0..400

Cf. A000522, A007526, A030297, A245334, A240993.

a111063 n = a111063_list !! n
a111063_list = 1 : zipWith (+) [1..] (zipWith (*) [0..] a111063_list)
-- Reinhard Zumkeller, Aug 30 2014

a:=proc(n) option remember; if n=0 then RETURN(1); fi; (n-1)*a(n-1)+n; end;
# Alternatively:
a := n -> `if`(n=0, 1, 2*exp(1)*GAMMA(n, 1) - 1):
seq(simplify(a(n)), n=0..22); # Peter Luschny, Nov 21 2017

FoldList[#1*#2 + #2 + 1 &, 1, Range[21]] (* Robert G. Wilson v, Jul 07 2012 *)

a(n+1) = Sum_{k=0..2*n} C(n,floor(k/2))(n-floor(k/2))!. - Paul Barry, May 04 2007
a(n) = A030297(n)/n, n>0.
a(n) = A007526(n) + A000522(n). - Gary Detlefs, Jun 10 2010
a(n) = 2*floor(e*n!) - 1, n>1. - Gary Detlefs, Jun 10 2010
E.g.f.: exp(x)*(1+x)/(1-x), - N. J. A. Sloane, May 03 2017
a(n) ~ 2*sqrt(2*Pi)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016
a(n) = 2*exp(1)*GAMMA(n, 1) - 1 for n>=1. - Peter Luschny, Nov 21 2017

A137948 Triangle read by rows, A000012 * A136579.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 4, 6, 5, 4, 6, 12, 24, 6, 5, 8, 18, 48, 120, 7, 6, 10, 24, 72, 240, 720, 8, 7, 12, 30, 96, 360, 1440, 5040, 9, 8, 14, 36, 120, 480, 2160, 10080, 40320, 10, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880

Offset: 0

Gary W. Adamson, Feb 28 2008

Row sums = A014144 starting (1, 3, 7, 17, 51, 205, ...).

T(n,k) = A245334(n,k) / A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Aug 31 2014

			First few rows of the triangle:
  1;
  2, 1;
  3, 2,  2;
  4, 3,  4,  6;
  5, 4,  6, 12, 24;
  6, 5,  8, 18, 48, 120;
  7, 6, 10, 24, 72, 240, 720;
  ...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A136579, A014144.

Cf. A245334, A007318.

a137948 n k = a137948_tabl !! n !! k
a137948_row n = a137948_tabl !! n
a137948_tabl = zipWith (zipWith div) a245334_tabl a007318_tabl
-- Reinhard Zumkeller, Aug 31 2014

As infinite lower triangular matrices, A000012 * A136579, where A000012 = (1; 1,1; 1,1,1; ...) and A136579 = (1; 1,1; 1,1,2; 1,1,2,6; 1,1,2,6,24; ...).

T(n,k) = (n+1-k) * k! for 0 <= k <= n. - Werner Schulte, Oct 06 2020

Offset changed by Reinhard Zumkeller, Aug 31 2014

A240993 A000142 (n+1) * A002109(n), a product of factorials and hyperfactorials.

Original entry on oeis.org

1, 2, 24, 2592, 3317760, 62208000000, 20316635136000000, 133852981198454784000000, 20211123400293732996612096000000, 78302033109811407811828935756349440000000, 8613223642079254859301182933198438400000000000000000

Offset: 0

Reinhard Zumkeller, Aug 31 2014

nonn

a(n+1) / a(n) = A055897(n+2);

row products of the triangle A245334.

Reinhard Zumkeller, Table of n, a(n) for n = 0..36

Cf. A000142, A002109, A245334, A055897, A245334, A111063, A074962.

Haskell
```
a240993 n = a000142 (n + 1) * a002109 n
```

Table[(n+1)!*Hyperfactorial[n], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[(n+1)*(n!)^(n+1)/BarnesG[n+1], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)

a(n) ~ A * sqrt(2*Pi) * n^(n^2/2+3*n/2+19/12) / exp(n*(n+4)/4), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

cp's OEIS Frontend

A049398 a(n) = (n+9)!/9!.

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A027810 a(n) = (n+1)*binomial(n+5, 5).

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A051431 a(n) = (n+10)!/10!.

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A070960 a(1) = 1; a(n) = n!*(3/2) for n>=2.

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A027818 a(n) = (n+1)*binomial(n+6,6).

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A047927 a(n) = n*(n-1)*(n-2)^2.

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A047927 a(n) = n(n-1)(n-2)^2.