A033312 -id:A033312 - cp's OEIS frontend

A173323 a(n) = 3*n! - 1.

2, 2, 5, 17, 71, 359, 2159, 15119, 120959, 1088639, 10886399, 119750399, 1437004799, 18681062399, 261534873599, 3923023103999, 62768369663999, 1067062284287999, 19207121117183999, 364935301226495999, 7298706024529919999, 153272826515128319999, 3372002183332823039999

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!-1: A033312 (k=1), A020543 (k=2), this sequence, A173321 (k=4), A173317 (k=5), A173316 (k=6).

[3*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

[2] cat [n eq 1 select n+1 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

Table[3 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)
3*(Range[0,20]!)-1 (* Harvey P. Dale, Oct 27 2015 *)

a(0)=2, a(n) = n*a(n-1)+n-1. - Vincenzo Librandi, Sep 30 2013
D-finite with recurrence a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Mar 07 2022
E.g.f.: 3/(1 - x) - exp(x). - Stefano Spezia, Oct 14 2024

A176487 Triangle read by rows: T(n,k) = binomial(n,k) + A008292(n+1,k+1) - 1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 71, 29, 1, 1, 61, 311, 311, 61, 1, 1, 125, 1205, 2435, 1205, 125, 1, 1, 253, 4313, 15653, 15653, 4313, 253, 1, 1, 509, 14635, 88289, 156259, 88289, 14635, 509, 1, 1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021, 1

Offset: 0

Roger L. Bagula, Apr 19 2010

			Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   13,    13,      1;
  1,   29,    71,     29,       1;
  1,   61,   311,    311,      61,       1;
  1,  125,  1205,   2435,    1205,     125,      1;
  1,  253,  4313,  15653,   15653,    4313,    253,     1;
  1,  509, 14635,  88289,  156259,   88289,  14635,   509,    1;
  1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021,   1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000182, A007318, A008292, A033312, A036563, A141689.

A176487:= func< n, k | Binomial(n, k) + EulerianNumber(n+1, k) - 1 >;
[A176487(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 31 2024

A176487 := proc(n,k)
    binomial(n,k)+A008292(n+1,k+1)-1 ;
end proc: # R. J. Mathar, Jun 16 2015

Needs["Combinatorica`"];
T[n_, k_, 0]:= Binomial[n, k];
T[n_, k_, 1]:= Eulerian[1 + n, k];
T[n_, k_, q_]:= T[n,k,q] = T[n,k,q-1] + T[n,k,q-2] - 1;
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten

# from sage.all import * # (use for Python)
from sage.combinat.combinat import eulerian_number
def A176487(n,k): return binomial(n,k) +eulerian_number(n+1,k) -1
print(flatten([[A176487(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 31 2024

T(n, k) = A007318(n,k) + A008292(n+1,k+1) - 1, 0 <= k <= n.
Sum_{k=0..n} T(n, k) = 2^n - n + A033312(n+1) (row sums).
T(n, k) = 2*A141689(n+1,k+1) - 1. - R. J. Mathar, Jan 19 2011
From G. C. Greubel, Dec 31 2024: (Start)
T(n, n-k) = T(n, k).
T(n, 1) = A036563(n+1).
Sum_{k=0..n} (-1)^k * T(n,k) = ((-1)^(n/2)*A000182(n/2 + 1) - 1)*(1 + (-1)^n)/2 + [n=0]. (End)

A179455 Triangle read by rows: number of permutation trees of power n and height <= k + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 6, 1, 15, 23, 24, 1, 52, 106, 119, 120, 1, 203, 568, 700, 719, 720, 1, 877, 3459, 4748, 5013, 5039, 5040, 1, 4140, 23544, 36403, 39812, 40285, 40319, 40320, 1, 21147, 176850, 310851, 354391, 362057, 362836, 362879, 362880

Offset: 0

Peter Luschny, Aug 11 2010

Partial row sums of A179454. Special cases: A179455(n,1) = BellNumber(n) = A000110(n) for n > 1; A179455(n,n-1) = n! for n > 1 and A179455(n,n-2) = A033312(n) for n > 1. Column 3 is A187761(n) for n >= 3.

See the interpretation of Joerg Arndt in A187761: Maps such that f^[k](x) = f^[k-1](x) correspond to column k of A179455 (for n >= k). - Peter Luschny, Jan 08 2013

			As a (0,0)-based triangle with an additional column [1,0,0,0,...] at the left hand side:
1;
0, 1;
0, 1, 2;
0, 1, 5, 6;
0, 1, 15, 23, 24;
0, 1, 52, 106, 119, 120;
0, 1, 203, 568, 700, 719, 720;
0, 1, 877, 3459, 4748, 5013, 5039, 5040;
0, 1, 4140, 23544, 36403, 39812, 40285, 40319, 40320;
0, 1, 21147, 176850, 310851, 354391, 362057, 362836, 362879, 362880;

Alois P. Heinz, Rows n = 0..141, flattened
Swapnil Garg, Alan Peng, Classical and consecutive pattern avoidance in rooted forests, arXiv:2005.08889 [math.CO], May 2020.
Peter Luschny, Permutation Trees.

Row sums are A264151.

Cf. A000110, A179454, A179456, A187761, A264428.

b[n_, t_, h_] := b[n, t, h] = If[n == 0 || h == 0, 1, Sum[Binomial[n - 1, j - 1]*b[j - 1, 0, h - 1]*b[n - j, t, h], {j, 1, n}]];
T[0, 0] = 1; T[n_, k_] := b[n, 1, k];
Table[T[n, k], {n, 0, 9}, {k, 0, If[n == 0, 0, n-1]}] // Flatten (* Jean-François Alcover, Jul 10 2019, after Alois P. Heinz in A179454 *)

# Generating algorithm from Joerg Arndt.
def A179455row(n):
    def generate(n, k):
        if n == 0 or k == 0: return 0
        for j in range(n-1, 0, -1):
            f = a[j] + 1
            while f <= j:
                a[j] = f1 = fl = f
                for i in range(k):
                    fl = f1
                    f1 = a[fl]
                if f1 == fl: return j
                f += 1
            a[j] = 0
        return 0
    count = [1 for j in range(n)] if n > 0 else [1]
    for k in range(n):
        a = [0 for j in range(n)]
        while generate(n, k) != 0:
            count[k] += 1
    return count
for n in range(9): A179455row(n) # Peter Luschny, Jan 08 2013

# uses[bell_transform from A264428]
# Adds the column (1,0,0,0,..) to the left hand side and starts at n=0.
def A179455_matrix(dim):
    b = [1]+[0]*(dim-1); L = [b]
    for k in range(dim):
        b = [sum(bell_transform(n, b)) for n in range(dim)]
        L.append(b)
    return matrix(ZZ, dim, lambda n, k: L[k][n] if k<=n else 0)
print(A179455_matrix(10)) # Peter Luschny, Dec 06 2015

A255341 Numbers n such that there is exactly one 1 in their factorial base representation (A007623).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 17, 19, 20, 23, 24, 28, 36, 40, 42, 46, 49, 50, 53, 54, 58, 61, 62, 65, 67, 68, 71, 73, 74, 77, 78, 82, 85, 86, 89, 91, 92, 95, 97, 98, 101, 102, 106, 109, 110, 113, 115, 116, 119, 120, 124, 132, 136, 138, 142, 168, 172, 180, 184, 186, 190, 192, 196, 204, 208, 210

Offset: 1

Antti Karttunen, Apr 27 2015

			The factorial base representation (A007623) of 5 is "21", which contains exactly one 1, thus 5 is included in the sequence.
The f.b.r. of 23 is "321", with only one 1, thus 23 is included in the sequence.
The f.b.r. of 24 is "1000", with only one 1, thus 24 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..13069

Cf. A007623, A257511, A255411, A255342, A255343.
Subsequence of A256450.
Subsequences: A000142, A033312 (apart from its zero-terms).

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 210}]; {0}~Join~Flatten@ Position[s, x_ /; DigitCount[x][[1]] == 1] (* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)

A068480 Numbers n such that gcd(n!-1,2^n+1)>1.

Original entry on oeis.org

1, 29, 41, 53, 69, 105, 125, 141, 153, 165, 189, 233, 249, 273, 293, 321, 329, 405, 413, 429, 441, 453, 485, 581, 585, 629, 641, 653, 713, 729, 741, 761, 765, 809, 813, 849, 893, 905, 989, 993, 1005, 1013, 1041, 1049, 1089, 1121, 1125, 1133, 1169, 1205

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000051 (2^n+1), A033312 (n!-1).

Cf. A068481, A068482, A068483.

Filtered([1..1230],n->Gcd(Factorial(n)-1,2^n+1)>1); # Muniru A Asiru, Oct 16 2018

select(n->gcd(factorial(n)-1,2^n+1)>1,[$1..1230]); # Muniru A Asiru, Oct 16 2018

Select[Range[2500], GCD[#! - 1, 2^# + 1] > 1 &] (* G. C. Greubel, Oct 15 2018 *)

isok(n) = gcd(n!-1, 2^n+1) > 1; \\ Michel Marcus, Oct 16 2018

A068483 Numbers n such that gcd(n!-1,2^n-1)>1.

Original entry on oeis.org

11, 15, 35, 75, 83, 111, 119, 135, 155, 179, 219, 231, 243, 323, 359, 375, 455, 459, 483, 491, 515, 519, 525, 531, 551, 611, 615, 639, 651, 663, 699, 719, 723, 735, 771, 779, 783, 803, 879, 915, 923, 939, 999, 1043, 1103, 1119, 1175, 1199, 1271, 1323

Offset: 1

Benoit Cloitre, Mar 10 2002

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000225 (2^n-1), A033312 (n!-1).

Cf. A068480, A068481, A068482.

Select[Range[2500], GCD[#! - 1, 2^# - 1] > 1 &] (* G. C. Greubel, Oct 15 2018 *)

isok(n) = gcd(n!-1,2^n-1) > 1; \\ Michel Marcus, Oct 16 2018

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 34, 33, 15, 5, 1, 1, 154, 153, 65, 23, 6, 1, 1, 874, 873, 339, 119, 32, 7, 1, 1, 5914, 5913, 2103, 719, 186, 42, 8, 1, 1, 46234, 46233, 15171, 5039, 1230, 267, 54, 9, 1, 1, 409114, 409113, 124755, 40319, 9258, 1891, 380

Offset: 0

Paul D. Hanna, Jan 04 2007

Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, ...;
1,(2), 3, (4), 5, 6, (7), 8, 9, 10, (11), 12, 13, 14, 15, (16), ...;
1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
Columns include:
k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
k=2: A007489 (Sum of k!, k=1..n);
k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
k=4: A033312 (n! - 1);
k=5: Partial sums of A001705;
k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).

Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.

{T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b-1)/2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

A207324 List of permutations of 1,2,3,...,n for n=1,2,3,..., in the order they are output by Steinhaus-Johnson-Trotter algorithm.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 2, 1, 2, 3, 1, 2, 1, 3, 1, 2, 3, 4, 1, 2, 4, 3, 1, 4, 2, 3, 4, 1, 2, 3, 4, 1, 3, 2, 1, 4, 3, 2, 1, 3, 4, 2, 1, 3, 2, 4, 3, 1, 2, 4, 3, 1, 4, 2, 3, 4, 1, 2, 4, 3, 1, 2, 4, 3, 2, 1, 3, 4, 2, 1, 3, 2, 4, 1, 3, 2, 1, 4

Offset: 1

R. J. Cano, Sep 14 2012

This table is otherwise similar to A030298, but lists permutations in the order given by the Steinhaus-Trotter-Johnson algorithm. - Antti Karttunen, Dec 28 2012

			For the set of the first two natural numbers {1,2} the unique permutations possible are 12 and 21, concatenated with 1 for {1} the resulting sequence would be 1, 1, 2, 2, 1.
If we consider up to 3 elements {1,2,3}, we have 123, 132, 312, 321, 231, 213 and the concatenation gives: 1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 2, 1, 2, 3, 1, 2, 1, 3.
Up to N concatenations, the sequence will have a total of Sum_{k=1..N} (k! * k) = (N+1)! - 1 = A033312(N+1) terms.

R. J. Cano, Table of n, a(n) for n = 1..10000
Joerg Arndt, C programs related to this sequence
R. J. Cano, Sequencer programs and additional information
Selmer M. Johnson, Generation of permutations by adjacent transposition, Mathematics of Computation, 17 (1963), p. 282-285.
Wikipedia, Steinhaus-Johnson-Trotter algorithm
Index entries for sequences related to permutations

Cf. A030298, A055881.
Cf. A001563 (row lengths), A001286 (row sums).
Pair (A130664(n),A084555(n)) = (1,1),(2,3),(4,5),(6,8),(9,11),(12,14),... gives the starting and ending offsets of the n-th permutation in this list.

Edited by N. J. A. Sloane, Antti Karttunen and R. J. Cano

A231722 Partial sums of A001088 starting from its second term; a(1)=0, a(n) = A001088(2)+...+A001088(n).

Original entry on oeis.org

0, 1, 3, 7, 23, 55, 247, 1015, 5623, 24055, 208375, 945655, 9793015, 62877175, 487550455, 3884936695, 58243116535, 384392195575, 6255075618295, 53220543000055, 616806151581175, 6252662237392375, 130241496125238775, 1122152167228009975, 20960365589283433975

Offset: 1

Antti Karttunen, Nov 27 2013

nonn

a(n+1) gives the index to the last term in each row of A231716. Specifically, for all n>=1, A231716(a(n+1)) = A033312(n+1).
a(n) = natural number which is written as the n-th repunit in "totient phi number system": 0, 1, 10, 11, 100, 101, 110, 111, 200, 201, 210, 211, 300, 301, 310, 311, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 1200, ..., and so on. Note how the 1st, the 3rd, the 7th and 23rd terms of this list are 1, 11, 111, and 1111.
In this number system the i-th digit from right (the least significant digit = digit_0) may contain integers in range 0..A000010(i+3)-1, and the value of the number is obtained as sum_{i=0..} digit_i * A001088(i+2).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

One less than A231721.

Cf. A000010 (Euler's totient function phi), A001088 (its partial products, "phitorials"), A231716, A033312.

with(numtheory): A231722:=n->add(product(phi(k), k=1..i), i=2..n): seq(A231722(n), n=1..20); # Wesley Ivan Hurt, Aug 09 2014

Table[Sum[Product[EulerPhi[k], {k, i}], {i, 2, n}], {n, 20}] (* Wesley Ivan Hurt, Aug 09 2014 *)

a(n) = sum(i=2, n, prod(k=1, i, eulerphi(k))); \\ Michel Marcus, Aug 09 2014

Scheme
```
(define (A231722 n) (- (A231721 n) 1))
```

a(n) = A231721(n)-1. [The terms are one less than the partial sums of "phitorials", A001088, cumulatively summed from their first term]

A275832 Size of the cycle containing element 1 in finite permutations listed in tables A060117 & A060118: a(n) = A007814(A275725(n)).

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 4, 1, 3, 1, 4, 2, 4, 2, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 5, 4, 5, 1, 4, 1, 5, 2, 5, 3, 4, 3, 5, 4, 5, 1, 2, 1, 4, 3, 4, 1, 2, 1, 5, 4, 5, 1, 3, 1, 5, 3, 5, 2, 3, 2, 5, 4, 5, 1, 3, 1, 4, 2, 4, 1, 3, 1, 5, 3, 5, 1, 4, 1, 5, 2, 5, 2, 4, 2, 5, 3, 5, 2, 3, 2, 4, 3, 4, 2, 3, 2, 5, 4, 5, 2, 4, 2, 5, 3, 5, 3, 4, 3, 5, 4, 5, 1

Offset: 0

Antti Karttunen, Aug 11 2016

nonn

			For n=0, the permutation with rank 0 in list A060118 is "1" (identity permutation) where 1 is fixed (in a 1-cycle), thus a(0)=1.
For n=1, the permutation with rank 1 in list A060118 is "21" where 1 is in a transposition (a 2-cycle), thus a(1)=2.
For n=3, the permutation with rank 3 in list A060118 is "231" where 1 is in a 3-cycle, thus a(3)=3.
For n=16, the permutation with rank 16 in list A060118 is "3412" (1 is in the other of two disjoint transpositions (1 3) and (2 4)), thus a(16)=2.
For n=44, the permutation with rank 44 in list A060118 is "43251", where 1 is a part of 3-cycle, thus a(44)=3.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A060117, A060118.
Cf. A000142, A007814, A033312, A275725, A275807.
Cf. A153880 (positions of 1's), A273670 (of terms larger than one), A275833 (of odd terms), A275834 (of even terms).

(define (A275832 n) (A007814 (A275725 n)))

a(n) = A007814(A275725(n)).
Other identities:
For n >= 1, a(A033312(n)) = n.
For n >= 2, a(A000142(n)) = 1.

cp's OEIS Frontend

A173323 a(n) = 3*n! - 1.

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A176487 Triangle read by rows: T(n,k) = binomial(n,k) + A008292(n+1,k+1) - 1.

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A179455 Triangle read by rows: number of permutation trees of power n and height <= k + 1.

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A255341 Numbers n such that there is exactly one 1 in their factorial base representation (A007623).

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A068480 Numbers n such that gcd(n!-1,2^n+1)>1.

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A068483 Numbers n such that gcd(n!-1,2^n-1)>1.

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PARI

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

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A207324 List of permutations of 1,2,3,...,n for n=1,2,3,..., in the order they are output by Steinhaus-Johnson-Trotter algorithm.

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