A001339 -id:A001339 - cp's OEIS frontend

A006183 a(n) = (n+1)a(n-1) + (2-n)a(n-2).

1, 2, 6, 22, 98, 522, 3262, 23486, 191802, 1753618, 17755382, 197282022, 2387112466, 31249472282, 440096734638, 6635304614542, 106638824162282, 1819969265702946, 32873194861759462, 626524419718239158, 12565295306571352002, 264532532769923200042

Offset: 1

N. J. A. Sloane, Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. C. Holroyd and W. J. G. Wingate, Cycles in the complement of a tree or other graph, Discrete Math., 55 (1985), 267-282.

Equals A030297(n-1) - A030297(n-2) + 1. Cf. A054096.

Equals 2 * A001339(n+2).

[n le 2 select n else n*Self(n-1)+(3-n)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 06 2016

RecurrenceTable[{a[n] == (n + 1) a[n - 1] + (2 - n) a[n - 2], a[0] == 1, a[1] == 2}, a, {n, 20}] (* Robert G. Wilson v, Jun 15 2013 *)

G.f.: 2*Sum_{k>=0} k!*(x/(1-x))^k - 1 = Q(0) -1, where Q(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013

More terms from James Sellers, Aug 21 2000
a(1) from Robert G. Wilson v, Jun 15 2013
a(21)-a(22) from Vincenzo Librandi, Mar 06 2016

A026243 a(n) = A000522(n) - 2.

Original entry on oeis.org

0, 3, 14, 63, 324, 1955, 13698, 109599, 986408, 9864099, 108505110, 1302061343, 16926797484, 236975164803, 3554627472074, 56874039553215, 966858672404688, 17403456103284419, 330665665962403998, 6613313319248079999, 138879579704209680020, 3055350753492612960483

Offset: 1

N. J. A. Sloane, based on a message from a correspondent who wishes to remain anonymous, Dec 21 2003

nonn

Number of operations of addition and multiplication needed to evaluate a determinant of order n by cofactor expansion.

			To calculate a determinant of order 3:
    |a b c|       |e f|       |d f|       |d e|
D = |d e f| = a * |h i| - b * |g i| + c * |g h| =
    |g h i|
   = a * (e*i - f*h) - b * (d*i - f*g) + c * (d*h - e*g).
There are 9 multiplications * and 5 additions (+ or -), so 14 operations and a(3) = 14. - _Bernard Schott_, Apr 21 2019

Alois P. Heinz, Table of n, a(n) for n = 1..170
C. Dubbs and D. Siegel, Computing determinants, College Math. J., 18 (1987), 48-49.
A. R. Pargeter, The vanishing coffee morning, Math. Gaz., 76 (1992), 386-387.
P. G. Sawtelle, The ubiquitous e, Math. Mag., 49 (1976), 244-245. [_N. J. A. Sloane_, Jan 29 2009]

Cf. A000522, A007526. Equals A033312 + A038156.

Cf. A001339.

a:= proc(n) a(n):= n*(a(n-1)+2)-1: end: a(1):= 0:
seq (a(n), n=1..30);  # Alois P. Heinz, May 25 2012

Table[E*Gamma[n+1, 1] - 2, {n, 1, 30}] (* Jean-François Alcover, May 18 2018 *)

a(n) = n*(a(n-1)+2)-1 for n>1, a(1) = 0. - Alois P. Heinz, May 25 2012
Conjecture: a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Jun 23 2013 [Confirmed by Altug Alkan, May 18 2018]
a(n) = floor(e*n!) - 2. - Bernard Schott, Apr 21 2019

A036918 a(n) = floor(e(n-1)(n-1)!).

Original entry on oeis.org

0, 2, 10, 48, 260, 1630, 11742, 95900, 876808, 8877690, 98641010, 1193556232, 15624736140, 220048367318, 3317652307270, 53319412081140, 909984632851472, 16436597430879730, 313262209859119578, 6282647653285676000, 132266266384961600020, 2916471173788403280462

Offset: 1

N. J. A. Sloane

nonn

Also the number of positive integers with all distinct digits expressed in base n. E.g., a(10) = Sum_{j=1..10} A073531(j). - Labos Elemer, Dec 05 2002

For example, for n=3 we have 1, 2, 10, 12, 20, 21, 102, 120, 201, 210 (10 numbers in total). - Igor Krasikov, Aug 14 2023

Cf. A001113, A073531, A073532.
a(n) = A001339(n)-1.
Equals (n-1)*A000522(n-1).

Table[Apply[Plus, Table[((b-1)/b)*Binomial[b, j]*j!, {j, 1, b}]], {b, 1, 25}]
Table[Floor[E(n-1)(n-1)!],{n,25}] (* Harvey P. Dale, May 19 2025 *)

G.f.: Q(0)/(2*x) - 1/x - 1/(1-x), where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013

a(n) = 2*(A038155(n) - A038155(n-1)). - Anton Zakharov, Oct 13 2016

A073591 a(n) = A000522(n) + 1.

Original entry on oeis.org

2, 3, 6, 17, 66, 327, 1958, 13701, 109602, 986411, 9864102, 108505113, 1302061346, 16926797487, 236975164806, 3554627472077, 56874039553218, 966858672404691, 17403456103284422, 330665665962404001, 6613313319248080002

Offset: 0

Vladeta Jovovic, Aug 28 2002

nonn

a(n) is an upper bound on the Ramsey numbers in A003323. - D. G. Rogers, Aug 27 2006

There is a nice derivation of the recurrence relation given in the Walker reference.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (28 terms from Vincenzo Librandi)
R. C. Walker, A graph coloring theorem, Math. Gaz., 60 (1976), 54-57.

Cf. A000522, A001339, A003323, A007526.

a:= proc(n) a(n):= `if`(n=0, 2, n*a(n-1)-n+2) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 17 2014

f[n_] := n*(f[n - 1] - 1) + 2;f[0]=2; ff[n_]:=(1/(1+n))(1+E*Gamma[1+n, 1]-E*(n^2)*Gamma[1+n, 1]+E*n*Gamma[2+n, 1]) (Spindler)
Table[FunctionExpand[Gamma[n, 1] E] + 1, {n, 2, 29}] (* Vincenzo Librandi, Feb 17 2014 *)

Conjecture: a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Feb 16 2014

a(n) = n*(a(n-1) - 1) + 2. - Georg Fischer, Dec 24 2023 [from the Walker reference, p. 55]

A095722 E.g.f.: exp(x)/(1-x)^8.

Original entry on oeis.org

1, 9, 89, 961, 11265, 142601, 1940089, 28245729, 438351041, 7226001865, 126122874201, 2324074591169, 45094140207169, 919088049256521, 19633713260950265, 438708172312264801, 10234490436580101249

Offset: 0

Philippe Deléham, Jul 08 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n) for x = 1, 2, 3, 4, 5, 6, 7 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..250

With[{nn=20},CoefficientList[Series[Exp[x]/(1-x)^8,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 26 2013 *)
Table[HypergeometricPFQ[{8, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*8^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+7)! / 7!.
a(n) = 2F0(8,-n;;-1). - Benedict W. J. Irwin, May 27 2016

A276589 Transpose of A276588.

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 49, 38, 30, 24, 261, 212, 174, 144, 120, 1631, 1370, 1158, 984, 840, 720, 11743, 10112, 8742, 7584, 6600, 5760, 5040, 95901, 84158, 74046, 65304, 57720, 51120, 45360, 40320, 876809, 780908, 696750, 622704, 557400, 499680, 448560, 403200, 362880, 8877691, 8000882, 7219974, 6523224, 5900520, 5343120, 4843440, 4394880, 3991680, 3628800

Offset: 0

Antti Karttunen, Sep 19 2016

Rows give the successive first differences of A001339.

			The top left corner of the array:
     1,     3,     11,      49,      261,      1631,      11743
     2,     8,     38,     212,     1370,     10112,      84158
     6,    30,    174,    1158,     8742,     74046,     696750
    24,   144,    984,    7584,    65304,    622704,    6523224
   120,   840,   6600,   57720,   557400,   5900520,   68019240
   720,  5760,  51120,  499680,  5343120,  62118720,  780827760
  5040, 45360, 448560, 4843440, 56775600, 718709040, 9778048560

Index entries for sequences related to factorial base representation

Topmost row: A001339. For other rows and columns, see the information given in transpose A276588.

Cf. also A276587.

T[r_, c_]:=Sum[Binomial[r, k](1 + c + k)!, {k, 0, r}]; Table[T[r - c, c], {r, 0, 10}, {c, 0, r}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)

T(r, c) = sum(k=0, r, binomial(r, k)*(1 + c + k)!);
for(r=0, 10, for(c=0, r, print1(T(r - c, c),", ");); print();) \\ Indranil Ghosh, Apr 11 2017

from sympy import binomial, factorial
def T(r, c): return sum([binomial(r, k) * factorial(1 + c + k) for k in range(r + 1)])
for r in range(11): print([T(r - c, c) for c in range(r + 1)]) # Indranil Ghosh, Apr 11 2017

(define (A276589 n) (A276588bi (A025581 n) (A002262 n))) ;; Code for A276588bi given in A276588.

A023805 Xenodromes: all digits in base 11 are different.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

Olivier Gérard

Considering some base b, there are b numbers with 1 digit, (b-1)*(b-1) numbers with 2 digits -- since leading 0's are not allowed and the second digit must avoid the first. There are (b-1)*(b-1)*(b-2) numbers with 3 digits, (b-1)*(b-1)*(b-2)*..*(b-d+1) numbers with d digits, in total b+(b-1)*sum_{d=2..b} (b-1)!/(b-d)! = b+(b-1)^2* 2F0(1,2-b;;-1) = A001339(b-1). The formula is applicable to sequences A023798 - A023810. This sequence here as A001339(11-1) = 98641011 terms. [From R. J. Mathar, Jan 27 2010]

Last term is a(98641011) = 282458553905. - Charles R Greathouse IV, Jun 16 2012

			121 (in decimal) = 100 (base 11) is a member of A168186 but not a member of this sequence. - Robert Munafo, Jan 26 2010
156 is in A023805 but not in A168186. - Franklin T. Adams-Watters, Jan 26 2010

Paolo Xausa, Table of n, a(n) for n = 1..10000

All three of A023805, A160453, A168186 are different.

Select[Range[0, 100], Max[DigitCount[#, 11]] == 1 &] (* Paolo Xausa, Mar 22 2025 *)

A051256 Numbers formed from binomial coefficients (mod 2) interpreted as digits in factorial base.

Original entry on oeis.org

1, 3, 7, 33, 121, 843, 5167, 46233, 362881, 3991683, 40279687, 522910113, 6227383801, 93409304523, 1313941673647, 22324392524313, 355687428096001, 6758061133824003, 122000787836928007, 2561305169719296033

Offset: 0

Antti Karttunen, Oct 24 1999

			a(5) = 1! + 2! + 5! + 6! = 843 (only the first, second, fifth and sixth terms are odd in row 5 of Pascal's Triangle).

Chai Wah Wu, Table of n, a(n) for n = 0..448

Cf. A001317, A001339, A048757, A047999.

A051256(n) := proc(n) local i; RETURN(add(((binomial(n,i) mod 2)*((i+1)!)),i=0..n)); end;

Table[Sum[(k+1)!Mod[Binomial[n,k],2],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Feb 14 2013 *)

from math import factorial
def A051256(n):
    return sum(0 if ~n & k else factorial(k+1) for k in range(n+1)) # Chai Wah Wu, Feb 08 2016

a(n) = Sum_{k=0..n} (k+1)!(C(n, k) mod 2).

A089900 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,...}.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 11, 4, 1, 120, 49, 18, 5, 1, 720, 261, 92, 27, 6, 1, 5040, 1631, 536, 159, 38, 7, 1, 40320, 11743, 3552, 1029, 256, 51, 8, 1, 362880, 95901, 26608, 7353, 1848, 389, 66, 9, 1, 3628800, 876809, 223456, 58095, 14384, 3125, 564, 83, 10, 1

Offset: 0

Paul D. Hanna, Nov 14 2003

Row 1 is A001339, antidiagonal sums form A089902 and the main diagonal is A089901; the next lower diagonal forms {1,4,27,256,..,n^n,..}, which is the hyperbinomial transform (cf. A088956) of the main diagonal.

			Note secondary diagonal: {(n+1)^(n+1)}; rows begin:
1, 2,. 6,. 24,. 120,.. 720,.. 5040,..
1, 3, 11,. 49,. 261,. 1631,. 11743,..
1,_4, 18,. 92,. 536,. 3552,. 26608,..
1, 5,_27, 159, 1029,. 7353,. 58095,..
1, 6, 38,_256, 1848, 14384, 121264,..
1, 7, 51, 389,_3125, 26595, 241015,..
1, 8, 66, 564, 5016,_46656, 456048,..
1, 9, 83, 787, 7701, 78077,_823543,..

Cf. A001339, A088956, A089901, A089902.

t[n_, k_] := (n^(k+2) - Exp[n]*(n-k-1)*Gamma[k+2, n])/(k+1) // Round; Table[t[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jun 24 2013 *)

T(n,k)=if(n<0 || k<0,0,sum(i=0,k,n^(k-i)*binomial(k,i)*(i+1)!))

T(0, k)=(k+1)!, T(n+1, n)=(n+1)^(n+1), T(n, k)=sum_{i=0..k}n^(k-i)*binomial(k, i)*(i+1)!

E.g.f.: 1/((1-y*exp(x))*(1-x)^2). E.g.f. (n-th row): exp(n*x)/(1-x)^2.

A095740 E.g.f.: exp(x)/(1-x)^9.

Original entry on oeis.org

1, 10, 109, 1288, 16417, 224686, 3288205, 51263164, 848456353, 14862109042, 274743964621, 5346258202000, 109249238631169, 2339328151461718, 52384307381414317, 1224472783033479556, 29826054965115774145

Offset: 0

Philippe Deléham Jul 09 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n), A095722(n) for x = 1, 2, 3, 4, 5, 6, 7, 8.

Robert Israel, Table of n, a(n) for n = 0..440

seq(simplify(hypergeom([9,-n],[],-1)),n=0..30); # Robert Israel, May 27 2016

Table[HypergeometricPFQ[{9, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*9^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+8)!/8!.
a(n) = 2F0(9,-n;;-1). - Benedict W. J. Irwin, May 27 2016
a(n) = ((n^8 + 28*n^7 + 350*n^6 + 2492*n^5 + 10899*n^4 + 29596*n^3 + 48082*n^2 + 42048*n + 14833) * Gamma(n+1,1)*e + n^7 + 28*n^6 + 349*n^5 + 2465*n^4 + 10579*n^3 + 27501*n^2 + 40132*n + 25487) / 40320. - Robert Israel, May 27 2016

cp's OEIS Frontend

A006183 a(n) = (n+1)*a(n-1) + (2-n)*a(n-2).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A026243 a(n) = A000522(n) - 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A036918 a(n) = floor(e*(n-1)*(n-1)!).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A073591 a(n) = A000522(n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A095722 E.g.f.: exp(x)/(1-x)^8.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A276589 Transpose of A276588.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

A023805 Xenodromes: all digits in base 11 are different.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A051256 Numbers formed from binomial coefficients (mod 2) interpreted as digits in factorial base.

Views

Author

A006183 a(n) = (n+1)a(n-1) + (2-n)a(n-2).

A036918 a(n) = floor(e(n-1)(n-1)!).