A276955 -id:A276955 - cp's OEIS frontend

A276949 Index of row where n is located in array A276953 (equally: column in A276955).

0, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 0

Antti Karttunen, Sep 22 2016

This is the smallest difference that occurs between any nonzero digit's radix (which is one more than its one-based position from the right) and that digit's value in the factorial base representation of n. See A225901 and the example.

a(0) = 0 by convention, as there are no nonzero digits present, and neither does 0 occur in arrays A276953 & A276955.

			For n=8, its factorial base representation (A007623) is "110", where the radix for each digit position 1, 2, 3 (from the right) is 2, 3, 4 (one larger than the position). For the 1 in the middle position the difference is 3-1 = 2, while for the 1 at the left we obtain 4-1 = 3. Of these two differences 2 is smaller, thus a(8)=2.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Index entries for sequences related to factorial base representation

Cf. A007623, A257679, A266193, A276950.
Cf. A276951 (for the other index).
Cf. arrays A276953 & A276955.
Cf. also A225901, A273667, A275847.

a(0) = 0, and for n >= 1: if A276950(n) = 1, then a(n) = 1, otherwise a(n) = 1 + a(A266193(n)).
Other identities. For all n >= 0:
a(n) = A257679(A225901(n)) = A257679(A275847(n)) = A257679(A273667(n)).

A276951 Index of column where n is located in array A276953 (equally: row in A276955).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 1, 5, 2, 6, 7, 8, 3, 9, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19, 5, 20, 21, 22, 2, 23, 6, 24, 25, 26, 7, 27, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 3, 37, 9, 38, 39, 40, 4, 41, 10, 42, 43, 44, 11, 45, 12, 46, 47, 48, 49, 50, 51, 52, 53, 54, 13, 55, 14, 56, 57, 58, 15, 59, 16, 60, 61, 62, 17, 63, 18, 64

Offset: 0

Antti Karttunen, Sep 22 2016

a(0) = 0 by convention, because 0 is not present in arrays A276953 and A276955.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Index entries for sequences related to factorial base representation

Cf. A260736, A266193, A273670, A276950, A276952, A273663.
Cf. arrays A276953 & A276955. A276949 gives their other index.
Cf. also A257681, A275847.

(definec (A276951 n) (cond ((zero? n) n) ((not (zero? (A276950 n))) (A276952 n)) (else (A276951 (A266193 n)))))

a(0) = 0; for n >= 1, if A260736(n) > 0 [when A276950(n) is not zero, when n is in A273670], then a(n) = A276952(n) = 1 + A273663(n), otherwise a(n) = a(A266193(n)).
Other identities. For all n >= 0:
a(n) = A257681(A275847(n)).

A276617 Square array A(n,k) = A276955(n,k)/k!, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 1, 3, 1, 4, 4, 1, 5, 6, 5, 1, 6, 8, 7, 7, 1, 7, 10, 9, 13, 9, 1, 8, 12, 11, 21, 16, 10, 1, 9, 14, 13, 31, 25, 18, 11, 1, 10, 16, 15, 43, 36, 28, 19, 13, 1, 11, 18, 17, 57, 49, 40, 29, 25, 15, 1, 12, 20, 19, 73, 64, 54, 41, 41, 28, 16, 1, 13, 22, 21, 91, 81, 70, 55, 61, 45, 30, 17, 1, 14, 24, 23, 111, 100, 88, 71, 85, 66, 48, 31, 18

Offset: 1

Antti Karttunen, Sep 22 2016

			The top left corner of the array:
1,  1,  1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1
3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17,  18,  19
4,  6,  8, 10, 12, 14, 16,  18,  20,  22,  24,  26,  28,  30,  32,  34,  36
5,  7,  9, 11, 13, 15, 17,  19,  21,  23,  25,  27,  29,  31,  33,  35,  37
7, 13, 21, 31, 43, 57, 73,  91, 111, 133, 157, 183, 211, 241, 273, 307, 343
9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361

Transpose: A276616.
Cf. A000142, A276955.
Columns 1-3: A273670, A276931, A276934.
Row 1: A000012, Row 2: n+2, Row 3: 2n+2, Row 4: 2n+3 (for n >= 1).
Row 5: A002061 (from a(3)=7 onward).
Row 6: squares (A000290, from a(3)=9 onward).
Row 7: A028552 (from a(2)=10 onward).
Row 8: A028387 (from a(2)=11 onward).

(define (A276617 n) (A276617bi (A002260 n) (A004736 n)))
(define (A276617bi row col) (/ (A276955bi row col) (A000142 col)))

A(n,k) = A276955(n,k)/k!

A276956 Inverse permutation to A276955.

Original entry on oeis.org

1, 2, 3, 6, 10, 4, 15, 5, 21, 28, 36, 9, 45, 14, 55, 66, 78, 91, 105, 120, 136, 153, 171, 7, 190, 20, 210, 231, 253, 8, 276, 27, 300, 325, 351, 35, 378, 44, 406, 435, 465, 496, 528, 561, 595, 630, 666, 13, 703, 54, 741, 780, 820, 19, 861, 65, 903, 946, 990, 77, 1035, 90, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 104

Offset: 1

Antti Karttunen, Sep 22 2016

Inverse: A276955.
Cf. A276949, A276951.
Related permutations: A257506, A275847.

(define (A276956 n) (let ((row (A276951 n)) (col (A276949 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A276949(n), and r = A276951(n).
As a composition of other permutations:
a(n) = A257506(A275847(n)).

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

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

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

			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

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).

T. D. Noe, Table of n, a(n) for n=1..100
Barry Balof and Helen Jenne, Tilings, Continued Fractions, Derangements, Scramblings, and e, Journal of Integer Sequences, Vol. 17 (2014), #14.2.7.
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math., Vol. 22, No. 1 (1970), pp. 22-35.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 97.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 641.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 101.
Helen K. Jenne, Proofs you can count on, Honors Thesis, Math. Dept., Whitman College, 2013.
B. D. Josephson and J. M. Boardman, Problems Drive 1961, Eureka, The Journal of the Archimedeans, Vol. 24 (1961), p. 20; entire volume.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Eric Weisstein's World of Mathematics, Uniform Sum Distribution.
Index entries for sequences related to factorial base representation
Index entries for sequences related to factorial numbers

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.

[Factorial(n)+Factorial(n+1): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
seq(n!+(n-1)!,n=1..25);
```

Table[n! + (n + 1)!, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 14 2011 *)
Total/@Partition[Range[0, 20]!, 2, 1] (* Harvey P. Dale, Nov 29 2013 *)

a(n)=denominator(polcoeff((x-1)*exp(x+x*O(x^(n+1))),n+1)); \\ Gerry Martens, Aug 12 2015

vector(30, n, (n+1)*(n-1)!) \\ Michel Marcus, Aug 12 2015

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

More terms from James Sellers, Sep 19 2000

A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

Original entry on oeis.org

0, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

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

For n >= 1 a(n) is the size of the centralizer of a transposition in the symmetric group S_(n+1). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
For n > 0, a(n) = n! - A062119(n-1) = number of permutations of length n that have two specified elements adjacent. For example, a(4) = 12 as of the 24 permutations, 12 have say 1 and 2 adjacent: 1234, 2134, 1243, 2143, 3124, 3214, 4123, 4213, 3412, 3421, 4312, 4321. - Jon Perry, Jun 08 2003
With different offset, denominators of certain sums computed by Ramanujan.
From Michael Somos, Mar 04 2004: (Start)
Stirling transform of a(n) = [2, 4, 12, 48, 240, ...] is A000629(n) = [2, 6, 26, 150, 1082, ...].
Stirling transform of a(n-1) = [1, 2, 4, 12, 48, ...] is A007047(n-1) = [1, 3, 11, 51, 299, ...].
Stirling transform of a(n) = [1, 4, 12, 48, 240, ...] is A002050(n) = [1, 5, 25, 149, 1081, ...].
Stirling transform of 2*A006252(n) = [2, 2, 4, 8, 28, ...] is a(n) = [2, 4, 12, 48, 240, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 2*A005649(n) = [4, 16, 88, 616, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 4*A083410(n) = [4, 16, 88, 616, ...]. (End)
Number of {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
Permanent of the (0, 1)-matrices with (i, j)-th entry equal to 0 if and only if it is in the border but not the corners. The border of a matrix is defined the be the first and the last row, together with the first and the last column. The corners of a matrix are the entries (i = 1, j = 1), (i = 1, j = n), (i = n, j = 1) and (i = n, j = n). - Simone Severini, Oct 17 2004

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 520.

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 490.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 817.
Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 11.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for sequences related to factorial base representation.

Essentially the same sequence as A098558.
Cf. A062119, A080046, A245334, A000142.
Row 3 of A276955 (from term a(2)=4 onward).

a052849 n = if n == 0 then 0 else 2 * a000142 n
a052849_list = 0 : fs where fs = 2 : zipWith (*) [2..] fs
-- Reinhard Zumkeller, Aug 31 2014

[0] cat [2*Factorial(n-1): n in [2..25]]; // Vincenzo Librandi, Nov 03 2014

spec := [S,{B=Cycle(Z),C=Cycle(Z),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0}, 2Range[20]!] (* Harvey P. Dale, Jul 13 2013 *)

PARI
```
a(n)=if(n<1,0,n!*2)
```

a(n) = T(n, 2) for n>1, where T is defined as in A080046.
D-finite with recurrence: {a(0) = 0, a(1) = 2, (-1 - n)*a(n+1) + a(n+2)=0}.
E.g.f.: 2*x/(1-x).
a(n) = A090802(n, n - 1) for n > 0. - Ross La Haye, Sep 26 2005
For n >= 1, a(n) = (n+3)!*Sum_{k=0..n+2} (-1)^k*binomial(2, k)/(n + 3 - k). - Milan Janjic, Dec 14 2008
G.f.: 2/Q(0) - 2, where Q(k) = 1 - x*(k + 1)/(1 - x*(k + 1)/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Apr 01 2013
G.f.: -2 + 2/Q(0), where Q(k) = 1 + k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: W(0) - 2 , where W(k) = 1 + 1/( 1 - x*(k+1)/( x*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n, n-1), n > 0. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=1} 1/a(n) = (e-1)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (e-1)/(2*e). (End)

More terms from Ross La Haye, Sep 26 2005

A052560 a(n) = 3*n!.

Original entry on oeis.org

3, 3, 6, 18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400, 1437004800, 18681062400, 261534873600, 3923023104000, 62768369664000, 1067062284288000, 19207121117184000, 364935301226496000, 7298706024529920000, 153272826515128320000

Offset: 0

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

a(n) is the size of the centralizer of a 3-cycle in the symmetric group S_(n+3). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001

3 times factorial numbers. - Omar E. Pol, Jan 17 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 502
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
Index entries for sequences related to factorial base representation

Cf. A000142, A129380.

Row 13 of A276955 (from term a(3)=18 onward).

[3*Factorial(n): n in [0..25]]; // Vincenzo Librandi, Jun 13 2011

spec := [S,{S=Union(Sequence(Z),Sequence(Z),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a:=n->sum(n!, k=1..n):seq(a(n)-sum(n!, k=4..n), n=0..19); # Zerinvary Lajos, Dec 22 2008

3*Range[0, 25]! (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *)

a(n)=3*n! \\ Charles R Greathouse IV, Nov 20 2011

[3*factorial(n) for n in range(25)] # G. C. Greubel, May 05 2019

E.g.f.: 3/(1-x).
a(n) = n*a(n-1), with a(0) = 3.
For n>0: a(n) = Sum_{k=1..n+1} A083746(k). - Reinhard Zumkeller, Apr 14 2007

A276945 Square array A(row,col): A(row,1) = A276155(row), and for col > 1, A(row,col) = A276154(A(row,col-1)); Dispersion of primorial base left shift A276154.

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 30, 36, 12, 5, 210, 240, 60, 14, 7, 2310, 2520, 420, 66, 32, 9, 30030, 32340, 4620, 450, 216, 38, 10, 510510, 540540, 60060, 4830, 2340, 246, 42, 11, 9699690, 10210200, 1021020, 62370, 30240, 2550, 270, 44, 13, 223092870, 232792560, 19399380, 1051050, 512820, 32550, 2730, 276, 62, 15

Offset: 1

Antti Karttunen, Sep 24 2016

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Entries in column k are all multiples of A002110(k-1). Dividing that factor out gives array A286625. - Antti Karttunen, Jun 30 2017

			The top left corner of the array:
   1,  2,   6,   30,   210,    2310,    30030,    510510
   3,  8,  36,  240,  2520,   32340,   540540,  10210200
   4, 12,  60,  420,  4620,   60060,  1021020,  19399380
   5, 14,  66,  450,  4830,   62370,  1051050,  19909890
   7, 32, 216, 2340, 30240,  512820,  9729720, 223603380
   9, 38, 246, 2550, 32550,  542850, 10240230, 233303070
  10, 42, 270, 2730, 34650,  570570, 10720710, 242492250
  11, 44, 276, 2760, 34860,  572880, 10750740, 243002760
  13, 62, 426, 4650, 60270, 1023330, 19429410, 446696250
  15, 68, 456, 4860, 62580, 1053360, 19939920, 456395940
  16, 72, 480, 5040, 64680, 1081080, 20420400, 465585120
  17, 74, 486, 5070, 64890, 1083390, 20450430, 466095630
  18, 90, 630, 6930, 90090, 1531530, 29099070, 669278610

Inverse permutation: A276946.
Transpose: A276943. One more than A286615.
Column 1: A276155.
Row 1: A002110.
Row 2: A276939.
Row 3: A088860 (2*A002110).
Row 11: 2*A276939 (row 2) from 16, 72, 480, 5040, 64680, ... onward.
Row 13: 3*A002110, from 18, 90, 630, 6930, 90090, ... onward.
Cf. A276154.
Cf. also arrays A286625, A276955.

(define (A276945 n) (A276945bi (A002260 n) (A004736 n)))
(define (A276945bi row col) (if (= 1 col) (A276155 row) (A276154 (A276945bi row (- col 1)))))

A(row,1) = A276155(row); for row > 1, A(row,col) = A276154(A(row,col-1)).

A276953 Square array A(row,col) read by antidiagonals: A(1,col) = A273670(col-1), and for row > 1, A(row,col) = A153880(A(row-1,col)); Dispersion of factorial base shift A153880 (array transposed).

Original entry on oeis.org

1, 3, 2, 4, 8, 6, 5, 12, 30, 24, 7, 14, 48, 144, 120, 9, 26, 54, 240, 840, 720, 10, 32, 126, 264, 1440, 5760, 5040, 11, 36, 150, 744, 1560, 10080, 45360, 40320, 13, 38, 168, 864, 5160, 10800, 80640, 403200, 362880, 15, 50, 174, 960, 5880, 41040, 85680, 725760, 3991680, 3628800, 16, 56, 246, 984, 6480, 46080, 367920, 766080, 7257600, 43545600, 39916800

Offset: 1

Antti Karttunen, Sep 22 2016

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Entries on row n are all multiples of n!. Dividing that factor out gives another array A276616.

			The top left corner of the array:
    1,    3,     4,     5,     7,     9,    10,    11,    13,    15,    16
    2,    8,    12,    14,    26,    32,    36,    38,    50,    56,    60
    6,   30,    48,    54,   126,   150,   168,   174,   246,   270,   288
   24,  144,   240,   264,   744,   864,   960,   984,  1464,  1584,  1680
  120,  840,  1440,  1560,  5160,  5880,  6480,  6600, 10200, 10920, 11520
  720, 5760, 10080, 10800, 41040, 46080, 50400, 51120, 81360, 86400, 90720

Inverse permutation: A276954.
Transpose: A276955.
Cf. A276949 (index of row where n appears), A276951 (index of column).
Cf. A153880, A273670.
Row 1: A273670, Row 2: A276932, Row 3: A276933.
Column 1: A000142. For other columns, see the rows of transposed array A276955.
Related or similar permutations: A257503, A275848, A273666.
Cf. also arrays A276616, A276589 & A276943.

(define (A276953 n) (A276953bi (A002260 n) (A004736 n)))
(define (A276953bi row col) (if (= 1 row) (A273670 (- col 1)) (A153880 (A276953bi (- row 1) col))))

A(1,col) = A273670(col-1), and for row > 1, A(row,col) = A153880(A(row-1,col))
As a composition of other permutations:
a(n) = A275848(A257503(n)).
Other identities. For all n >= 1:
A(A276949(n),A276951(n)) = n.

A052649 Expansion of e.g.f. (2+x-x^2)/(1-x)^2.

Original entry on oeis.org

2, 5, 14, 54, 264, 1560, 10800, 85680, 766080, 7620480, 83462400, 997920000, 12933043200, 180583603200, 2702527027200, 43153254144000, 732297646080000, 13160434839552000, 249692574523392000, 4987449116762112000, 104614786351595520000, 2299092397726924800000

Offset: 0

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

a(1) is 5 and gives the row number in the table of 0-origin permutations of order 3 in which the first 3 items are reversed. Row 5 of this table is 2 1 0. a(2) is 14 and gives the row number in the table of 0-origin permutations of order 4 in which the first three items are reversed. Row 14 of this table is 2 1 0 3.... a(6) is 10800 and gives the row number in the table of 0-origin permutations of order 8 in which the first 3 items are reversed. Row 10800 of this table is 2 1 0 3 4 5 6 7. Et cetera. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004

In factorial base representation (A007623) the terms of this sequence are written as: 10, 21, 210, 2100, 21000, 210000, ... From a(1) = 5 = "21" onward each term begins always with "21", which is then followed by n-1 zeros. - Antti Karttunen, Sep 24 2016

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 596.
Index entries for sequences related to factorial base representation.

Cf. A000142, A001048, A007623, A129326.

Row 4 of A276955 (from a(1)=5 onward).

spec := [S,{S=Prod(Sequence(Z),Union(Z,Sequence(Z),Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

f[n_] := (3 + 2 n) n!; f[0] = 2; Array[f, 19, 0]
a[n_] := a[n] = a[n - 1]*n (2 n + 3)/(2 n + 1); a[0] = 2; a[1] = 5; Array[ a, 19, 0] (* Robert G. Wilson v *)
With[{nn=20},CoefficientList[Series[(2+x-x^2)/(1-x)^2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 09 2017 *)

a(n)=if(n<=1,[2,5][n+1], a(n-1)*(n*(2*n+3))/(2*n+1) );
for(n=0,11,print1(a(n),", "))

(define (A052649 n) (if (zero? n) 2 (+ (A000142 n) (* 2 (A000142 (+ 1 n)))))) ;; Antti Karttunen, Sep 24 2016

a(n) = (3+2*n)*n!.
E.g.f.: -(-x+x^2-2)/(-1+x)^2.
Recurrence: a(0)=2, a(1)=5, (-7*n-5-2*n^2)*a(n)+(3+2*n)*a(n+1)=0 for n>=1.
a(n) = A129326(n), n>1. - R. J. Mathar, Jun 14 2008
a(n) = (n+1)*a(n-1) - 2*A001048(n-1). - Gary Detlefs, Dec 16 2009
a(0) = 2; for n >= 1, a(n) = 2*(n+1)! + n! - Antti Karttunen, Sep 24 2016
From Amiram Eldar, Feb 17 2024: (Start)
Sum_{n>=0} 1/a(n) = 1/6 + e/2 - erfi(1)*sqrt(Pi)/4, where erfi is the imaginary error function.
Sum_{n>=0} (-1)^n/a(n) = 1/6 - 1/(2*e) + erf(1)*sqrt(Pi)/4, where erf is the error function. (End)

cp's OEIS Frontend

A276949 Index of row where n is located in array A276953 (equally: column in A276955).

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A276951 Index of column where n is located in array A276953 (equally: row in A276955).

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A276617 Square array A(n,k) = A276955(n,k)/k!, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A276956 Inverse permutation to A276955.

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

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A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

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A052560 a(n) = 3*n!.

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