cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.

Original entry on oeis.org

0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880, 19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200, 22376988058521600, 431565146817638400, 8752948036761600000, 186244810780170240000
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Number of permutations of n+1 elements with exactly two cycles.
Number of cycles in all permutations of [n]. Example: a(3) = 11 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) have 11 cycles altogether. - Emeric Deutsch, Aug 12 2004
Row sums of A094310: In the symmetric group S_n, each permutation factors into k independent cycles; a(n) = sum k over S_n. - Harley Flanders (harley(AT)umich.edu), Jun 28 2004
The sum of the top levels of the last column over all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, the levels of their last columns being 2 and 1, respectively. - Emeric Deutsch, Aug 12 2006
a(n) is divisible by n for all composite n >= 6. a(2*n) is divisible by 2*n + 1. - Leroy Quet, May 20 2007
For n >= 2 the determinant of the n-1 X n-1 matrix M(i,j) = i + 2 for i = j and 1 otherwise (i,j = 1..n-1). E.g., for n = 3 the determinant of [(3, 1), (1, 4)]. See 53rd Putnam Examination, 1992, Problem B5. - Franz Vrabec, Jan 13 2008, Mar 26 2008
The numerator of the fraction when we sum (without simplification) the terms in the harmonic sequence. (1 + 1/2 = 2/2 + 1/2 = 3/2; 3/2 + 1/3 = 9/6 + 2/6 = 11/6; 11/6 + 1/4 = 44/24 + 6/24 = 50/24;...). The denominator of this fraction is n!*A000142. - Eric Desbiaux, Jan 07 2009
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=1) ~ exp(-x)/x^2*(1 - 3/x + 11/x^2 - 50/x^3 + 274/x^4 - 1764/x^5 + 13068/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
a(n) is the number of permutations of [n+1] containing exactly 2 cycles. Example: a(2) = 3 because the permutations (1)(23), (12)(3), (13)(2) are the only permutations of [3] with exactly 2 cycles. - Tom Woodward (twoodward(AT)macalester.edu), Nov 12 2009
It appears that, with the exception of n= 4, a(n) mod n = 0 if n is composite and = n-1 if n is prime. - Gary Detlefs, Sep 11 2010
a(n) is a multiple of A025527(n). - Charles R Greathouse IV, Oct 16 2012
Numerator of harmonic number H(n) = Sum_{i=1..n} 1/i when not reduced. See A001008 (Wolstenholme numbers) for the reduced numerators. - Rahul Jha, Feb 18 2015
The Stirling transform of this sequence is A222058(n) (Harmonic-geometric numbers). - Anton Zakharov, Aug 07 2016
a(n) is the (n-1)-st elementary symmetric function of the first n numbers. - Anton Zakharov, Nov 02 2016
The n-th iterated integral of log(x) is x^n * (n! * log(x) - a(n))/(n!)^2 + a polynomial of degree n-1 with arbitrary coefficients. This can be proven using the recurrence relation a(n) = (n-1)! + n*a(n-1). - Mohsen Maesumi, Oct 31 2018
Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019
Total number of left-to-right maxima (or minima) in all permutations of [n]. a(3) = 11 = 3+2+2+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21. - Alois P. Heinz, Aug 01 2020

Examples

			(1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...
G.f. = x + x^2 + 5*x^3 + 14*x^4 + 94*x^5 + 444*x^6 + 3828*x^7 + 25584*x^8 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identities 186-190.
  • N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, 1986, see page 2. MR0863284 (89d:41049)
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
  • Shanzhen Gao, Permutations with Restricted Structure (in preparation).
  • K. Javorszky, Natural Orders: De Ordinibus Naturalibus, 2016, ISBN 978-3-99057-139-2.
  • 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).

Links

Crossrefs

Cf. A000399, A000454, A000482, A001233, A001234, A243569, A243570.
Cf. A000774, A004041, A024167, A046674, A049034, A008275.
Cf. A021009, A081358, A092691, A094587, A151881, A121633.
With signs: A081048.
Column 1 in triangle A008969.
Row sums of A136662.

Programs

  • Magma
    a:=[]; for n in [1..22] do a:=a cat [Abs(StirlingFirst(n,2))]; end for; a; // Marius A. Burtea, Jan 01 2020
  • Maple
    A000254 := proc(n) option remember; if n<=1 then n else n*A000254(n-1)+(n-1)!; fi; end: seq(A000254(n),n=0..21);
a := n -> add(n!/k, k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jan 22 2008
  • Mathematica
    Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] (* Wouter Meeussen *)
Table[ n!*HarmonicNumber[n], {n, 0, 19}] (* Robert G. Wilson v, May 21 2005 *)
Table[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}],{n,1,30}] (* Alexander Adamchuk, Jul 11 2006 *)
Abs[StirlingS1[Range[20],2]] (* Harvey P. Dale, Aug 16 2011 *)
Table[Gamma'[n + 1] /. EulerGamma -> 0, {n, 0, 30}] (* Li Han, Feb 14 2024*)
  • Maxima
    a(n):=(-1)^(n+1)/2*(n+1)*sum(k*bern(k-1)*stirling1(n,k),k,1,n); /* Vladimir Kruchinin, Nov 20 2016 */
  • MuPAD
    A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) := 1:
  • PARI
    {a(n) = if( n<0, 0, (n+1)! / 2 * sum( k=1, n, 1 / k / (n+1-k)))} /* Michael Somos, Feb 05 2004 */
  • Sage
    [stirling_number1(i, 2) for i in range(1, 22)]  # Zerinvary Lajos, Jun 27 2008

Formula

Let P(n,X) = (X+1)*(X+2)*(X+3)*...*(X+n); then a(n) is the coefficient of X; or a(n) = P'(n,0). - Benoit Cloitre, May 09 2002
Sum_{k > 0} a(k) * x^k/ k!^2 = exp(x) *(Sum_{k>0} (-1)^(k+1) * x^k / (k * k!)). - Michael Somos, Mar 24 2004; corrected by Warren D. Smith, Feb 12 2006
a(n) is the coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/2.
a(n) = n! * Sum_{i=1..n} 1/i = n! * H(n), where H(n) = A001008(n)/A002805(n) is the n-th harmonic number.
a(n) ~ 2^(1/2)*Pi^(1/2)*log(n)*n^(1/2)*e^-n*n^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: log(1 - x) / (x-1). (= (log(1 - x))^2 / 2 if offset 1). - Michael Somos, Feb 05 2004
D-finite with recurrence: a(n) = a(n-1) * (2*n - 1) - a(n-2) * (n - 1)^2, if n > 1. - Michael Somos, Mar 24 2004
a(n) = A081358(n)+A092691(n). - Emeric Deutsch, Aug 12 2004
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic, Jan 29 2005
p^2 divides a(p-1) for prime p > 3. a(n) = (Sum_{i=1..n} 1/i) / Product_{i=1..n} 1/i. - Alexander Adamchuk, Jul 11 2006
a(n) = 3* A001710(n) + 2* A001711(n-3) for n > 2; e.g., 11 = 3*3 + 2*1, 50 = 3*12 + 2*7, 274 = 3*60 + 2*47, ... - Gary Detlefs, May 24 2010
a(n) = A138772(n+1) - A159324(n). - Gary Detlefs, Jul 05 2010
a(n) = A121633(n) + A002672(n). - Gary Detlefs, Jul 18 2010
a(n+1) = Sum_{i=1..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 14 2010
From Gary Detlefs, Sep 11 2010: (Start)
a(n) = (a(n-1)*(n^2 - 2*n + 1) + (n + 1)!)/(n - 1) for n > 2.
It appears that, with the exception of n = 2, (a(n+1)^2 - a(n)^2) mod n^2 = 0 if n is composite and 4*n if n is prime.
It appears that, with the exception of n = 2, (a(n+1)^3 - a(n)^2) mod n = 0 if n is composite and n - 2 if n is prime.
It appears that, with the exception of n = 2, (a(n)^2 + a(n+1)^2) mod n = 0 if n is composite and = 2 if n is prime. (End)
a(n) = Integral_{x=0..oo} (x^n - n!)*log(x)*exp(-x) dx. - Groux Roland, Mar 28 2011
a(n) = 3*n!/2 + 2*(n-2)!*Sum_{k=0..n-3} binomial(k+2,2)/(n-2-k) for n >= 2. - Gary Detlefs, Sep 02 2011
a(n)/(n-1)! = ml(n) = n*ml(n-1)/(n-1) + 1 for n > 1, where ml(n) is the average number of random draws from an n-set with replacement until the total set has been observed. G.f. of ml: x*(1 - log(1 - x))/(1 - x)^2. - Paul Weisenhorn, Nov 18 2011
a(n) = det(|S(i+2, j+1)|, 1 <= i,j <= n-2), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
E.g.f.: x/(1 - x)*E(0)/2, where E(k) = 2 + E(k+1)*x*(k + 1)/(k + 2). - Sergei N. Gladkovskii, Jun 01 2013 [Edited by Michael Somos, Nov 28 2013]
0 = a(n) * (a(n+4) - 6*a(n+3) + 7*a(n+2) - a(n+1)) - a(n+1) * (4*a(n+3) - 6*a(n+2) + a(n+1)) + 3*a(n+2)^2 unless n=0. - Michael Somos, Nov 28 2013
For a simple way to calculate the sequence, multiply n! by the integral from 0 to 1 of (1 - x^n)/(1 - x) dx. - Rahul Jha, Feb 18 2015
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Inverse binomial transform of A073596.
a(n) ~ sqrt(2*Pi*n) * n^n * (log(n) + gamma)/exp(n), where gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = ((-1)^(n+1)/2*(n+1))*Sum_{k=1..n} k*Bernoulli(k-1)*Stirling1(n,k). - Vladimir Kruchinin, Nov 20 2016
a(n) = (n)! * (digamma(n+1) + gamma), where gamma is the Euler-Mascheroni constant A001620. - Pedro Caceres, Mar 10 2018
From Andy Nicol, Oct 21 2021: (Start)
Gamma'(x) = a(x-1) - (x-1)!*gamma, where Gamma'(x) is the derivative of the gamma function at positive integers and gamma is the Euler-Mascheroni constant. E.g.:
Gamma'(1) = -gamma, Gamma'(2) = 1-gamma, Gamma'(3) = 3-2*gamma,
Gamma'(22) = 186244810780170240000 - 51090942171709440000*gamma. (End)
From Peter Bala, Feb 03 2022: (Start)
The following are all conjectural:
E.g.f.: for nonzero m, (1/m)*Sum_{n >= 1} (-1)^(n+1)*(1/n)*binomial(m*n,n)* x^n/(1 - x)^(m*n+1) = x + 3*x^2/2! + 11*x^3/3! + 50*x^4/4! + ....
For nonzero m, a(n) = (1/m)*n!*Sum_{k = 1..n} (-1)^(k+1)*(1/k)*binomial(m*k,k)* binomial(n+(m-1)*k,n-k).
a(n)^2 = (1/2)*n!^2*Sum_{k = 1..n} (-1)^(k+1)*(1/k^2)*binomial(n,k)* binomial(n+k,k). (End)
From Mélika Tebni, Jun 20 2022: (Start)
a(n) = -Sum_{k=0..n} k!*A021009(n, k+1).
a(n) = Sum_{k=0..n} k!*A094587(n, k+1). (End)
a(n) = n! * 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n - 1)^2/((2*n - 1)))))). - Peter Bala, Mar 16 2024

A185105 Number T(n,k) of entries in the k-th cycles of all permutations of {1,2,..,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Original entry on oeis.org

1, 3, 1, 12, 5, 1, 60, 27, 8, 1, 360, 168, 59, 12, 1, 2520, 1200, 463, 119, 17, 1, 20160, 9720, 3978, 1177, 221, 23, 1, 181440, 88200, 37566, 12217, 2724, 382, 30, 1, 1814400, 887040, 388728, 135302, 34009, 5780, 622, 38, 1, 19958400, 9797760, 4385592, 1606446, 441383, 86029, 11378, 964, 47, 1
Offset: 1

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Author

Wouter Meeussen, Dec 26 2012

Keywords

Comments

Row sums are n!*n = A001563(n) (see example).
For fixed k>=1, A185105(n,k) ~ n!*n/2^k. - Vaclav Kotesovec, Apr 25 2017

Examples

			The six permutations of n=3 in ordered cycle form are:
{ {1}, {2}, {3}    }
{ {1}, {2, 3}, {}  }
{ {1, 2}, {3}, {}  }
{ {1, 2, 3}, {}, {}}
{ {1, 3, 2}, {}, {}}
{ {1, 3}, {2}, {}  }
.
The lengths of the cycles in position k=1 sum to 12, those of the cycles in position k=2 sum to 5 and those of the cycles in position k=3 sum to 1.
Triangle begins:
       1;
       3,     1;
      12,     5,     1;
      60,    27,     8,     1;
     360,   168,    59,    12,    1;
    2520,  1200,   463,   119,   17,   1;
   20160,  9720,  3978,  1177,  221,  23,  1;
  181440, 88200, 37566, 12217, 2724, 382, 30, 1;
  ...

Links

Crossrefs

Columns k=1-10 give: A001710(n+1), A138772, A159324(n-1)/2 or A285231, A285232, A285233, A285234, A285235, A285236, A285237, A285238.
T(2n,n) gives A285239.
Cf. A000142, A001563, A270236, A284816, A285439.

Programs

  • Maple
    b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+coeff(p, x, 0)*j*x^i)(b(n-j, i+1))*
       binomial(n-1, j-1)*(j-1)!, j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Apr 15 2017
  • Mathematica
    Table[it = Join[RotateRight /@ ToCycles[#], Table[{}, {k}]] & /@ Permutations[Range[n]]; Tr[Length[Part[#, k]]& /@ it], {n, 7}, {k, n}]
(* Second program: *)
b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[Function[p, p + Coefficient[ p, x, 0]*j*x^i][b[n-j, i+1]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]];
Array[T, 12] // Flatten (* Jean-François Alcover, May 30 2018, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Apr 15 2017

A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 4, 6, 0, 2, 16, 20, 24, 0, 14, 44, 80, 108, 120, 0, 90, 200, 384, 544, 672, 720, 0, 646, 1288, 2240, 3264, 4128, 4800, 5040, 0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320, 0, 47622, 78652, 123456, 176832, 231936, 280512, 323520, 352800, 362880
Offset: 0

Views

Author

Alois P. Heinz, Apr 02 2020

Keywords

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = n! for k>=n.

Examples

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    0,    2;
  0,    0,    4,     6;
  0,    2,   16,    20,    24;
  0,   14,   44,    80,   108,   120;
  0,   90,  200,   384,   544,   672,   720;
  0,  646, 1288,  2240,  3264,  4128,  4800,  5040;
  0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320;
  ...

Links

Crossrefs

Columns k=0-10 (for n>=k) give: A000007, A002464, A110128, A117574, A189255, A189256, A189271, A360384, A360386, A360462, A360463.
Main diagonal gives A000142.
T(2n,n) gives A189849.
T(n+1,n) gives 4*A138772(n).
T(n+2,n) gives 16*A333804(n).
Cf. A000170 (condition is satisfied for all k), A248686 (p(i) at distance k are sorted).

A159324 n! times the average number of comparisons required by an insertion sort of n (distinct) elements.

Original entry on oeis.org

0, 0, 2, 16, 118, 926, 7956, 75132, 777456, 8771184, 107307360, 1416252960, 20068629120, 304002322560, 4903642679040, 83928856838400, 1519397749094400, 29010025797580800, 582647327132774400, 12280347845905305600, 271030782903552000000, 6251213902855219200000
Offset: 0

Views

Author

Harmen Wassenaar (towr(AT)ai.rug.nl), Apr 10 2009

Keywords

Examples

			For n=3, insertion sorting 123, 213, 213, 231, 312, 321 takes 3+3+3+2+3+2 = 4*3+2*2 = 16 comparisons.

Links

Crossrefs

Row sums of triangle A159323.
Cf. A000254, A001008, A002805, A138772, A212395, A285231.

Programs

  • Maple
    a:= proc(n) option remember;
      `if`(n<2, 0, a(n-1)*n + (n-1)! * (n-1)*(n+2)/2)
    end:
seq(a(n), n=0..30); # Alois P. Heinz, May 14 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 2][n+1],
      ((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)))
    end:
seq(a(n), n=0..30); # Alois P. Heinz, Dec 16 2016
  • Mathematica
    a[n_] := n! ((n+1)(n+2)/4 - HarmonicNumber[n] - 1/2); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 12 2017, after Gary Detlefs *)

Formula

a(n) = a(n-1)*(n) + n! *(n+1)/2 - (n-1)!.
a(n) = Sum_k A159323(n,k) = Sum_k A129178(n,k) * (n(n-1)/2 - k).
a(n) = n!/4 *(n^2+3*n-4*H(n)), where H(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Sep 02 2010
a(n) = A138772(n+1) - A000254(n). - Gary Detlefs, May 13 2012
a(n) = ((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)) for n>2. - Alois P. Heinz, Dec 16 2016
a(n) = 2 * A285231(n+1). - Alois P. Heinz, Apr 15 2017

A138771 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 11, 5, 2, 24, 50, 26, 14, 6, 120, 274, 154, 94, 54, 24, 720, 1764, 1044, 684, 444, 264, 120, 5040, 13068, 8028, 5508, 3828, 2568, 1560, 720, 40320, 109584, 69264, 49104, 35664, 25584, 17520, 10800, 5040
Offset: 1

Views

Author

Emeric Deutsch, Apr 10 2008

Keywords

Comments

T(n,0)=(n-1)!=A000142(n-1).
T(n,1)=A000254(n-1).
T(n,2)=A001705(n-2).
T(n,3)=2*A001711(n-4).
T(n,4)=6*A001716(n-5).
T(n,n-1)=(n-2)! (n>=2).
Sum(kT(n,k),k=0..n-1)=(n-1)!(n-1)(n+2)/4=A138772(n).

Examples

			T(4,2)=5 because we have (1)(23)(4), (1)(24)(3), (13)(24), (12)(34) and (14)(23).
Triangle starts;
1;
1,1;
2,3,1;
6,11,5,2;
24,50,26,14,6;
120,274,154,94,54,24;

Crossrefs

Cf. A000142, A000254, A001705, A001711, A001716, A138772.
From Johannes W. Meijer, Oct 16 2009: (Start)
A000142 equals for n=>1 the row sums.
a(n) = A165680(n) * A165675(n-1).
(End)

Programs

  • Maple
    T:=proc (n,k) if k = 0 then factorial(n-1) elif n <= k then 0 else (n-1)*T(n-1, k)+factorial(n-2) end if end proc: for n to 9 do seq(T(n, k), k=0..n-1) end do;

Formula

T(n,k)=(n-1)T(n-1,k)+(n-2)! (1<=k<=n-1). The row generating polynomials P[n](t) satisfy: P[n+1](t)=nP[n](t)+(n-1)!(t+t^2+...+t^n).

A226167 Array read by antidiagonals: a(i,j) is the number of ways of labeling a tableau of shape (i,1^j) with the integers 1, 2, ... i+j-2 (each label being used once) such that the first row is decreasing, and the first column has m-1 labels.

Original entry on oeis.org

1, 3, 1, 12, 5, 1, 60, 27, 7, 1, 360, 168, 48, 9, 1, 2520, 1200, 360, 75, 11, 1, 20160, 9720, 3000, 660, 108, 13, 1, 181440, 88200, 27720, 6300, 1092, 147, 15, 1, 1814400, 887040, 282240, 65520, 11760, 1680, 192, 17, 1, 19958400, 9797760, 3144960, 740880, 136080, 20160, 2448, 243, 19, 1
Offset: 1

Views

Author

John M. Campbell, May 29 2013

Keywords

Comments

For an arbitrary composition c, let F_c^p denote the linear transformation of NSym that is adjoint to multiplication by the fundamental quasi-symmetric function indexed by c. Then a(i,j) equals the coefficient of H_(1,1) in (F_(1)^p)^(i+j-2)(H_(i,1^j)) (see below SAGE program, and Corollary 2.7 in the below link).
Let M(n) = [a(i,j)]_{n x n}. Then det(M(n))=A000178(n)=the n-th superfactorial.
Let p_n(x) denote the polynomial such that a(x,n)=p_n(x). Then the coefficient of x in p_n(x) is |A009575(n)|. For example, p_4(x)=4x^3+18x^2+26x+12, and the coefficient of x in p_4(x) is |A009575(4)|=26.
First row is A001710. Second row is A138772. Fourth row is A136659.

Examples

			There are a(3,2) = 7 ways of labeling the tableau of shape (3,1,1) with 1, 2 and 3 (with each label being used once) such that the first row is decreasing and the first column has 1 label:
1    2    3    X    X    X    X
X    X    X    1    2    3    X
X32  X31  X21  X32  X31  X21  321
The matrix [a(i,j)]_(6 x 6) is given below:
[1  3  12   60   360   2520]
[1  5  27  168  1200   9720]
[1  7  48  360  3000  27720]
[1  9  75  660  6300  65520]
[1 11 108 1092 11760 136080]
[1 13 147 1680 20160 257040]

Links

Crossrefs

Cf. A000178, A009575.
Main diagonal gives: A023999. - Alois P. Heinz, Jan 21 2014

Programs

  • Maple
    a:= (i, j)-> (i+j-2)!/i!*(2*i+j-1)*j/2:
seq(seq(a(i, 1+d-i), i=1..d), d=1..12);  # Alois P. Heinz, Jan 21 2014
  • Mathematica
    a[n_,k_]:=(n+k-2)!/n!*(2*n+k-1)*k/2 ;
Print[Array[a[#1,#2]&,{50,50}]//MatrixForm]
(* A program which gives a list of tableaux *)
a[i_, j_] :=  Module[{f, list1, el, emptylist, n},
  f[q_] := StringReplace[StringReplace[StringReplace[    StringReplace[ToString[q], ToString[i + j - 1] -> "X"], ", " -> ""], "{" -> ""], "}" -> ""]; list1 = Permutations[Join[Table[q, {q, 1, i + j - 2}], {i + j - 1, i + j - 1}]]; el[q_] := First[Take[list1, {q, q}]]; emptylist = {}; n = 1; While[n < 1 + Length[list1], If[Take[el[n], {j + 1, i + j}] == Sort[Take[el[n], {j + 1, i + j}], Greater] && Count[Take[el[n], {1, j + 1}], i + j - 1] == 2, emptylist = Append[emptylist, f[el[n]]], Null]; n++]; Print[emptylist]]
  • Sage
    NSym = NonCommutativeSymmetricFunctions(QQ) ;
QSym = QuasiSymmetricFunctions(QQ) ;
F = QSym.Fundamental() ;
H = NSym.complete() ;
def a(n, m):
     expr = H([n]+[1 for q in range(m)]) ;
     w=1 ;
     while w

Formula

a(i,j) = (i+j-2)!/i!*(2*i+j-1)*j/2.

A308498 Triangle read by rows where T(n,k), n>=1, 1<=k<=n is the number of (0,1)-matrices of size n with the first row and column sum = k and remaining sums = 1.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 24, 27, 10, 1, 120, 168, 88, 17, 1, 720, 1200, 800, 225, 26, 1, 5040, 9720, 7800, 2850, 486, 37, 1, 40320, 88200, 82320, 36750, 8232, 931, 50, 1, 362880, 887040, 940800, 493920, 136416, 20384, 1632, 65, 1, 3628800, 9797760, 11612160, 6985440
Offset: 1

Views

Author

Lars Blomberg, Jun 01 2019

Keywords

Examples

			For n=4, k=3:
1110  1101  1011  1110  1101  1011  1110  1101  1011  0111
1000  1000  1000  1000  1000  1000  0001  0010  0100  1000
1000  1000  1000  0001  0010  0100  1000  1000  1000  1000
0001  0010  0100  1000  1000  1000  1000  1000  1000  1000
so T(4,3)=10.
Triangle begins:
1
2,1
6,5,1
24,27,10,1
120,168,88,17,1
720,1200,800,225,26,1
5040,9720,7800,2850,486,37,1
40320,88200,82320,36750,8232,931,50,1
362880,887040,940800,493920,136416,20384,1632,65,1
3628800,9797760,11612160,6985440,2286144,423360,44928,2673,82,1
39916800,117936000,154224000,104328000,39372480,8678880,1144800,90450,4150,101,1

Crossrefs

Cf. A000142, A002522, A138772.

Formula

T(n,k) = ((n-1)!)^2 * (k^2+n-k) / ((k!)^2 * (n-k)!).
T(n,1) = A000142(n).
T(n,2) = A138772(n).
T(n,n-1) = A002522(n-1).
T(n,n) = 1.
Showing 1-7 of 7 results.

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