A052572 -id:A052572 - cp's OEIS frontend

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

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

Offset: 1

Antti Karttunen, Sep 22 2016

The square 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.

When viewed in factorial base (A007623) the terms on each row start all with the same prefix, but with an increasing number of zeros appended to the end. For example, for row 8 (A001344 from a(1)=11 onward), the terms written in factorial base look as: 121, 1210, 12100, 121000, ...

			The top left {1..9} x {1..18} corner of the array:
   1,  2,   6,   24,   120,    720,    5040,    40320,    362880
   3,  8,  30,  144,   840,   5760,   45360,   403200,   3991680
   4, 12,  48,  240,  1440,  10080,   80640,   725760,   7257600
   5, 14,  54,  264,  1560,  10800,   85680,   766080,   7620480
   7, 26, 126,  744,  5160,  41040,  367920,  3669120,  40279680
   9, 32, 150,  864,  5880,  46080,  408240,  4032000,  43908480
  10, 36, 168,  960,  6480,  50400,  443520,  4354560,  47174400
  11, 38, 174,  984,  6600,  51120,  448560,  4394880,  47537280
  13, 50, 246, 1464, 10200,  81360,  730800,  7297920,  80196480
  15, 56, 270, 1584, 10920,  86400,  771120,  7660800,  83825280
  16, 60, 288, 1680, 11520,  90720,  806400,  7983360,  87091200
  17, 62, 294, 1704, 11640,  91440,  811440,  8023680,  87454080
  18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400
  19, 74, 366, 2184, 15240, 121680, 1093680, 10926720, 120113280
  20, 78, 384, 2280, 15840, 126000, 1128960, 11249280, 123379200
  21, 80, 390, 2304, 15960, 126720, 1134000, 11289600, 123742080
  22, 84, 408, 2400, 16560, 131040, 1169280, 11612160, 127008000
  23, 86, 414, 2424, 16680, 131760, 1174320, 11652480, 127370880

Inverse permutation: A276956.
Transpose: A276953.
Cf. A276949 (index of column where n appears), A276951 (index of row).
Cf. A153880.
Columns 1-3: A273670, A276932, A276933.
The following lists some of the rows that have their own entries. Pattern present in the factorial base expansion of the terms on that row is given in double quotes:
Row 1: A000142 (from a(1)=1, "1" onward),
Row 2: A001048 (from a(2)=3, "11" onward),
Row 3: A052849 (from a(2)=4, "20" onward).
Row 4: A052649 (from a(1)=5, "21" onward).
Row 5: A108217 (from a(3)=7, "101" onward).
Row 6: A054119 (from a(3)=9, "111" onward).
Row 7: A052572 (from a(2)=10, "120" onward).
Row 8: A001344 (from a(1)=11, "121" onward).
Row 13: A052560 (from a(3)=18, "300" onward).
Row 16: A225658 (from a(1)=21, "311" onward).
Row 20: A276940 (from a(3) = 27, "1011" onward).
Related or similar permutations: A257505, A275848, A273666.
Cf. also arrays A276617, A276588 & A276945.

(define (A276955 n) (A276955bi (A002260 n) (A004736 n)))
(define (A276955bi row col) (if (= 1 col) (A273670 (- row 1)) (A153880 (A276955bi row (- col 1)))))

A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1))
As a composition of other permutations:
a(n) = A275848(A257505(n)).

A038720 a(n) = (n+3)*n!/2.

Original entry on oeis.org

2, 5, 18, 84, 480, 3240, 25200, 221760, 2177280, 23587200, 279417600, 3592512000, 49816166400, 741015475200, 11769069312000, 198766503936000, 3556874280960000, 67224923910144000, 1338096104497152000, 27978373094031360000, 613091306060513280000

Offset: 1

N. J. A. Sloane, May 02 2000

Next-to-last diagonal of A038719.

a(n-1) is the sum of the n-th entries in all cycles of all permutations of [n]. a(2) = 5 because the sum of the third entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 3+2+0+0+0+0 = 5. - Alois P. Heinz, May 03 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
Index entries for sequences related to posets.

Cf. A038719, A038721, A052572, A214178.

Main diagonal of A285793.

import Data.List (transpose)
a038720 n = a038720_list !! (n-1)
a038720_list = (transpose $ map reverse a038719_tabl) !! 1
-- Reinhard Zumkeller, Jul 08 2012

A038720:= func< n | (n+3)*Factorial(n)/2 >; // G. C. Greubel, May 11 2025

Array[(# + 3) #!/2 &, 21] (* Michael De Vlieger, Apr 28 2022 *)

def A038720(n): return (n+3)*factorial(n)//2 # G. C. Greubel, May 11 2025

a(n) = A052572(n)/2.
a(n) = A214178(n+3,n). - Reinhard Zumkeller, Jul 08 2012
G.f.: Sum_{n>=1} ( (n+1)*x/(1 + (n+1)*x) )^n. - Paul D. Hanna, Jan 02 2013
E.g.f.: 1/(1-x) + 1/(2*(x-1)^2) - 3/2. - Alois P. Heinz, May 04 2017
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*e - 14/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 10/e - 10/3. (End)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 09 2000.

A282466 a(n) = n*a(n-1) + n!, with n>0, a(0)=5.

Original entry on oeis.org

5, 6, 14, 48, 216, 1200, 7920, 60480, 524160, 5080320, 54432000, 638668800, 8143027200, 112086374400, 1656387532800, 26153487360000, 439378587648000, 7825123418112000, 147254595231744000, 2919482409811968000, 60822550204416000000, 1328364496464445440000

Offset: 0

Bruno Berselli, Feb 22 2017

C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), page 240 (Example 9.57 gives the recurrence).

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A229039.

Cf. sequences with formula (n + k)*n!: A052521 (k=-5), A282822 (k=-4), A052520 (k=-3), A052571 (k=-2), A062119 (k=-1), A001563 (k=0), A000142 (k=1), A001048 (k=2), A052572 (k=3), A052644 (k=4), this sequence (k=5).

A282466:= func< n | (n+5)*Factorial(n) >; // G. C. Greubel, May 14 2025

RecurrenceTable[{a[0] == 5, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}] (* or *)
Table[(n + 5) n!, {n, 0, 30}]

def A282466(n): return (n+5)*factorial(n) # G. C. Greubel, May 14 2025

E.g.f.: (5 - 4*x)/(1 - x)^2.
a(n) = (n + 5)*n!.
a(n) = 2*A229039(n) for n>0.
From Amiram Eldar, Nov 06 2020: (Start)
Sum_{n>=0} 1/a(n) = 9*e - 24.
Sum_{n>=0} (-1)^n/a(n) = 24 - 65/e. (End)

A185263 Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 15, 8, 1, 35, 84, 1, 70, 469, 180, 1, 126, 1869, 3044, 1, 210, 5985, 26060, 8064, 1, 330, 16401, 152900, 193248, 1, 495, 39963, 696905, 2286636, 604800, 1, 715, 88803, 2641925, 18128396, 19056960, 1, 1001, 183183, 8691683, 109425316, 292271616, 68428800, 1, 1365, 355355, 25537655, 539651112, 2961802480, 2699672832

Offset: 0

N. J. A. Sloane, Jan 21 2012

Row n contains floor(n/2) + 1 terms.

			Triangle begins:
  n\k| 0    1      2       3         4         5        6
-----+---------------------------------------------------
   0 | 1
   1 | 1
   2 | 1    1
   3 | 1    5
   4 | 1   15      8
   5 | 1   35     84
   6 | 1   70    469     180
   7 | 1  126   1869    3044
   8 | 1  210   5985   26060      8064
   9 | 1  330  16401  152900    193248
  10 | 1  495  39963  696905   2286636    604800
  11 | 1  715  88803 2641925  18128396  19056960
  12 | 1 1001 183183 8691683 109425316 292271616 68428800
  ...
Polynomials p_n(t):
  p_0 = t;
  p_1 = t^2;
  p_2 = t^3 +     t;
  p_3 = t^4 +   5*t^2;
  p_4 = t^5 +  15*t^3 +    8*t;
  p_5 = t^6 +  35*t^4 +   84*t^2;
  p_6 = t^7 +  70*t^5 +  469*t^3 +  180*t;
  p_7 = t^8 + 126*t^6 + 1869*t^4 + 3044*t^2;
  ...
A(x;t) = t + t^2*x/1! + (t^3 + t)*x^2/2! + (t^4 + 5*t^2)*x^3/3! + ...

Gheorghe Coserea, Rows n = 0..202, flattened
Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.

Cf. A038720, A048994, A060593, A185259.

For uncompressed form of polynomial coefficients, in order of increasing powers, see A164652.

T[n_, k_] := Abs[StirlingS1[n+2, n-2k+1]]/Binomial[n+2, 2];
Table[T[n, k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

seq(N) = {
  my(p=vector(N), t='t, v); p[1] = t^2; p[2] = t^3 + t;
  for (n=3, N,
    p[n] = ((2*n+1)*t*p[n-1] + (n-1)*(n^2-t^2)*p[n-2])/(n+2));
  v = vector(#p, n, vector(1+n\2, k, polcoeff(p[n], n+1-2*(k-1))));
  concat([[1]], v);
};
concat(seq(13))

N=14; x='x+O('x^(N+1));
concat(apply(p->select(a->a!=0, Vec(p)), Vec(serlaplace(((1-x)^(-t) - (1+x)^t)/x^2))))

T(n,k) = -stirling(n+2, n+1-2*k, 1)/binomial(n+2,2);
concat(1, concat(vector(13, n, vector(1+n\2, k, T(n, k-1)))))
\\ Gheorghe Coserea, Jan 29 2018

From Gheorghe Coserea, Jan 29 2018: (Start)
p(n) = Sum_{k=0..floor(n/2)} T(n,k)*t^(n+1-2*k) satisfies (n+2)*p(n) = (2*n+1)*t*p(n-1) + (n-1)*(n^2-t^2)*p(n-2), n >= 2. (th. 3, (iii))
E.g.f. A(x;t) = Sum_{n>=0} p(n)*x^n/n! = ((1-x)^(-t) - (1+x)^t)/x^2. (th. 3, (i))
T(n,k) = -Stirling1(n+2, n+1-2*k)/binomial(n+2,2), where Stirling1(n,k) is defined by A048994.
A000142(n) = p(n)(1), A052572(n) = p(n)(2) for n > 0, A060593(n) = T(2*n, n) for n > 0. (End)
n-th row polynomial R(n,x) satisfies x*R(n,x^2) = (1/2)*( P(n+1,x) - P(n+1,-x) )/ binomial(n+2,2), where P(k,x) = (1 + x)*(1 + 2*x) * ... *(1 + k*x). - Peter Bala, May 14 2023

More terms from Gheorghe Coserea, Jan 29 2018

cp's OEIS Frontend

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

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A038720 a(n) = (n+3)*n!/2.

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A282466 a(n) = n*a(n-1) + n!, with n>0, a(0)=5.

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A185263 Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.

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