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.

A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

Original entry on oeis.org

1, 1, -1, 2, -4, 1, 6, -18, 9, -1, 24, -96, 72, -16, 1, 120, -600, 600, -200, 25, -1, 720, -4320, 5400, -2400, 450, -36, 1, 5040, -35280, 52920, -29400, 7350, -882, 49, -1, 40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1, 362880, -3265920
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In absolute values, this sequence also gives the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals (j-1)^2 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006
A partial permutation on a set X is a bijection between two subsets of X. |T(n,n-k)| equals the numbers of partial permutations of an n-set having domain cardinality equal to k. Let E denote the operator D*x*D, where D is the derivative operator d/dx. Then E^n = Sum_{k = 0..n} |T(n,k)|*x^k*D^(n+k). - Peter Bala, Oct 28 2008
The unsigned triangle is the generalized Riordan array (exp(x), x) with respect to the sequence n!^2 as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!*(n+1)! is A105278). - Peter Bala, Aug 15 2013
The unsigned triangle appears on page 83 of Ser (1933). - N. J. A. Sloane, Jan 16 2020

Examples

			The triangle a(n,m) starts:
n\m   0       1      2       3      4      5    6  7  8
0:    1
1:    1      -1
2:    2      -4      1
3:    6     -18      9      -1
4:   24     -96     72     -16      1
5:  120    -600    600    -200     25     -1
6:  720   -4320   5400   -2400    450    -36    1
7: 5040  -35280  52920  -29400   7350   -882   49  -1
8:40320 -322560 564480 -376320 117600 -18816 1568 -64 1
...
From _Wolfdieter Lang_, Jan 31 2013 (Start)
Recurrence (usual one): a(4,1) = 7*(-18) - 6 - 3^2*(-4) = -96.
Recurrence (simplified version): a(4,1) = 5*(-18) - 6 = -96.
Recurrence (Sage program): |a(4,1)| = 6 + 3*18 + 4*9 = 96. (End)
Embedded recurrence (Maple program): a(4,1) = -4!*(1 + 3) = -96.

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. 799.
  • G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
  • J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.

Links

Crossrefs

Row sums give A009940, alternating row sums are A002720.
Column sequences (unsigned): A000142, A001563, A001809-A001812 for m=0..5.
Central terms: A295383.
For generators and generalizations see A132440.
Cf. A021010, A025166, A025167, A062137, A062138, A062139, A062140, A066667, A321620.

Programs

  • Magma
    /* As triangle: */ [[((-1)^k)*Factorial(n)*Binomial(n, k)/Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 18 2020
  • Maple
    A021009 := proc(n,k) local S; S := proc(n,k) option remember; `if`(k = 0, 1, `if`( k > n, 0, S(n-1,k-1)/k + S(n-1,k))) end: (-1)^k*n!*S(n,k) end: seq(seq(A021009(n,k), k=0..n), n=0..8); # Peter Luschny, Jun 21 2017
# Alternative for the unsigned case (function RiordanSquare defined in A321620):
RiordanSquare(add(x^m, m=0..10), 10, true); # Peter Luschny, Dec 06 2018
  • Mathematica
    Flatten[ Table[ CoefficientList[ n!*LaguerreL[n, x], x], {n, 0, 9}]] (* Jean-François Alcover, Dec 13 2011 *)
  • PARI
    p(n) = denominator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));
concat(1, concat(vector(9, n, Vec(-p(n)))))  \\ Gheorghe Coserea, Dec 01 2016
  • PARI
    {T(n, k) = if( n<0, 0, n! * polcoeff( sum(i=0, n, binomial(n, n-i) * (-x)^i / i!), k))}; /* Michael Somos, Dec 01 2016 */
  • PARI
    row(n) = Vecrev(n!*pollaguerre(n)); \\ Michel Marcus, Feb 06 2021
  • Sage
    def A021009_triangle(dim): # computes unsigned T(n,k).
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(2*k+1)*M[n-1,k]+(k+1)^2*M[n-1,k+1]
    return M
A021009_triangle(9) # Peter Luschny, Sep 19 2012

Formula

a(n, m) = ((-1)^m)*n!*binomial(n, m)/m! = ((-1)^m)*((n!/m!)^2)/(n-m)! if n >= m, otherwise 0.
E.g.f. for m-th column: (-x/(1-x))^m /((1-x)*m!), m >= 0.
Representation (of unsigned a(n, m)) as special values of Gauss hypergeometric function 2F1, in Maple notation: n!*(-1)^m*hypergeom([ -m, n+1 ], [ 1 ], 1)/m!. - Karol A. Penson, Oct 02 2003
Sum_{m>=0} (-1)^m*a(n, m) = A002720(n). - Philippe Deléham, Mar 10 2004
E.g.f.: (1/(1-x))*exp(x*y/(x-1)). - Vladeta Jovovic, Apr 07 2005
Sum_{n>=0, m>=0} a(n, m)*(x^n/n!^2)*y^m = exp(x)*BesselJ(0, 2*sqrt(x*y)). - Vladeta Jovovic, Apr 07 2005
Matrix square yields the identity matrix: L^2 = I. - Paul D. Hanna, Nov 22 2008
From Tom Copeland, Oct 20 2012: (Start)
Symbolically, with D=d/dx and LN(n,x)=n!L_n(x), define :Dx:^j = D^j x^j, :xD:^j = x^j D^j, and LN(.,x)^j = LN(j,x) = row polynomials of A021009.
Then some useful relations are
1) (:Dx:)^n = LN(n,-:xD:) [Rodriguez formula]
2) (xDx)^n = x^n D^n x^n = x^n LN(n,-:xD:) [See Al-Salam ref./A132440]
3) (DxD)^n = D^n x^n D^n = LN(n,-:xD:) D^n [See ref. in A132440]
4) umbral composition LN(n,LN(.,x))= x^n [See Rota ref.]
5) umbral comp. LN(n,-:Dx:) = LN(n,-LN(.,-:xD:)) = 2^n LN(n,-:xD:/2)= n! * (n-th row e.g.f.(x) of A038207 with x replaced by :xD:).
An example for 2) is the operator (xDx)^2 = (xDx)(xDx) = xD(x^2 + x^3D)= 2x^2 + 4x^3 D + x^4 D^2 = x^2 (2 + 4x D + x^2 D^2) = x^2 (2 + 4 :xD: + :xD:^2) = x^2 LN(2,-:xD:) = x^2 2! L_2(-:xD:).
An example of the umbral composition in 5) is given in A038207.
The op. xDx is related to the Euler/binomial transformation for power series/o.g.f.s. through exp(t*xDx) f(x) = f[x/(1-t*x)]/(1-t*x) and to the special Moebius/linear fractional/projective transformation z exp(-t*zDz)(1/z)f(z) = f(z/(1+t*z)).
For a general discussion of umbral calculus see the Gessel link. (End)
From Wolfdieter Lang, Jan 31 2013: (Start)
Standard recurrence derived from the three term recurrence of the orthogonal polynomials system {n!*L(n,x)}: L(n,x) = (2*n - 1 - x)*L(n-1,x) - (n-1)^2*L(n-2,x), n>=1, L(-1,x) = 0, L(0,x) = 1.
a(n,m) = (2*n-1)*a(n-1,m) - a(n-1,m-1) - (n-1)^2*a(n-2,m),
n >=1, with a(n,-1) = 0, a(0,0) = 1, a(n,m) = 0 if n < m. (compare this with Peter Luschny's program for the unsigned case |a(n,m)| = (-1)^m*a(n,m)).
Simplified recurrence (using column recurrence from explicit form for a(n,m) given above):
a(n,m) = (n+m)*a(n-1,m) - a(n-1,m-1), n >= 1, a(0,0) = 1, a(n,-1) = 0, a(n,m) = 0 if n < m. (End)
|T(n,k)| = [x^k] (-1)^n*U(-n,1,-x), where U(a,b,x) is Kummer's hypergeometric U function. - Peter Luschny, Apr 11 2015
T(n,k) = (-1)^k*n!*S(n,k) where S(n,k) is recursively defined by: "if k = 0 then 1 else if k > n then 0 else S(n-1,k-1)/k + S(n-1,k)". - Peter Luschny, Jun 21 2017
The unsigned case is the exponential Riordan square (see A321620) of the factorial numbers. - Peter Luschny, Dec 06 2018
Omitting the diagonal and signs, this array is generated by the commutator [D^n,x^n] = D^n x^n - x^n D^n = Sum_{i=0..n-1} ((n!/i!)^2/(n-i)!) x^i D^i on p. 9 of both papers by Belov-Kanel and Kontsevich. - Tom Copeland, Jan 23 2020

Extensions

Name changed and table given by Wolfdieter Lang, Nov 28 2011

A144084 T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 72, 96, 24, 1, 25, 200, 600, 600, 120, 1, 36, 450, 2400, 5400, 4320, 720, 1, 49, 882, 7350, 29400, 52920, 35280, 5040, 1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320
Offset: 0

Views

Author

Abdullahi Umar, Sep 10 2008, Sep 30 2008

Keywords

Comments

T(n,k) is also the number of elements in the Green's J equivalence classes in the symmetric inverse monoid, I sub n.
T(n,k) is also the number of ways to place k nonattacking rooks on an n X n chessboard. It can be obtained by performing P(n,k) permutations of n-columns over each C(n,k) combination of n-rows for the given k-rooks. The rule is also applicable for unequal (m X n) sized rectangular boards. - Antal Pinter, Nov 12 2014
Rows also give the coefficients of the matching-generating polynomial of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Rows also give the coefficients of the independence polynomial of the n X n rook graph and clique polynomial of the n X n rook complement graph. - Eric W. Weisstein, Jun 13 and Sep 14 2017
T(n,k) is the number of increasing subsequences of length n-k over all permutations of [n]. - Geoffrey Critzer, Jan 08 2023

Examples

			T(3,1) = 9 because there are exactly 9 partial bijections (on a 3-element set) of height 1, namely: (1)->(1), (1)->(2), (1)->(3), (2)->(1), (2)->(2), (2)->(3), (3)->(1), (3)->(2), (3)->(3).
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   2;
  1,  9,  18,    6;
  1, 16,  72,   96,   24;
  1, 25, 200,  600,  600,  120;
  1, 36, 450, 2400, 5400, 4320, 720;
  ...

References

  • O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, page 61.
  • J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995).
  • Vaclav Kotesovec, Non-attacking chess pieces, 6th ed. (2013), p. 216, p. 218.

Links

Crossrefs

T(n,k) = |A021010|. Sum of rows of T(n,k) is A002720. T(n,n) is the order of the symmetric group on an n-element set, n!.
Columns k=2..10 are A163102, A179058, A179059, A179060, A179061, A179062, A179063, A179064, A179065.
Cf. A089231, A105219, A295383.

Programs

  • Magma
    /* As triangle */ [[(Binomial(n,k)^2)*Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jun 13 2017
  • Maple
    T:= (n, k)-> (binomial(n, k)^2)*k!:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 04 2012
  • Mathematica
    Table[Table[Binomial[n, k]^2 k!,{k, 0, n}], {n, 0, 6}] // Flatten (* Geoffrey Critzer, Dec 04 2012 *)
Table[ CoefficientList[n!*LaguerreL[n, x], x] // Abs // Reverse, {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 18 2013 *)
CoefficientList[Table[n! x^n LaguerreL[n, -1/x], {n, 0, 8}], x] // Flatten (* Eric W. Weisstein, Apr 24 2017 *)
CoefficientList[Table[(-x)^n HypergeometricU[-n, 1, -(1/x)], {n, 5}],
  x] // Flatten (* Eric W. Weisstein, Jun 13 2017 *)
  • PARI
    T(n,k) = k! * binomial(n,k)^2 \\ Andrew Howroyd, Feb 13 2018

Formula

T(n,k) = (C(n,k)^2)*k!.
T(n,k) = A007318(n,k) * A008279(n,k). - Antal Pinter, Nov 12 2014
From Peter Bala, Jul 04 2016: (Start)
G.f.: exp(x*t)*I_0(2*sqrt(x)) = 1 + (1 + t)*x/1!^2 + (1 + 4*t + 2*t^2)*x^2/2!^2 + (1 + 9*t + 18*t^2 + 6*t^3)*x^3/3!^2 + ..., where I_0(x) = Sum_{n >= 0} (x/2)^(2*n)/n!^2 is a modified Bessel function of the first kind.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*n!/k!*binomial(n,k) *t^(n-k)*R(k,t). Cf. A089231. (End)
From Peter Bala, Oct 05 2019: (Start)
E.g.f.: 1/(1 - t*x)*exp(x/(1 - t*x)).
Recurrence for row polynomials: R(n+1,t) = (1 + (2*n+1)*t)R(n,t) - n^2*t^2*R(n-1,t), with R(0,t) = 1 and R(1,t) = 1 + t.
R(n,t) equals the denominator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The numerator polynomial is the (n+1)-th row polynomial of A089231. (End)
Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/A001044(n) = exp(y*x)*E(x) where E(x) = Sum_{n>=0} x^n/A001044(n). - Geoffrey Critzer, Jan 08 2023
Sum_{k=0..n} k*T(n,k) = A105219(n). - Alois P. Heinz, Jan 08 2023
T(n,k) = Sum_{d=0..2*k} c(k,d)*n^d, where c(k,d) = Sum_{j=max(d-k,0)..k} binomial(k,j)*A008275(k+j,d)/j!. - Eder G. Santos, Jan 23 2025

A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

Original entry on oeis.org

1, 2, 2, 9, 12, 6, 64, 96, 72, 24, 625, 1000, 900, 480, 120, 7776, 12960, 12960, 8640, 3600, 720, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 43046721
Offset: 1

Views

Author

Christian G. Bower, Dec 14 2001

Keywords

Comments

T(n,k) = number of endofunctions with k recurrent elements. - Mitch Harris, Jul 06 2006
The sum of row n is n^n, for any n. Basically the same sequence arises when studying random mappings (see A243203, A243202). - Stanislav Sykora, Jun 01 2014

Examples

			Triangle T(n,k) begins:
       1;
       2,      2;
       9,     12,      6;
      64,     96,     72,     24;
     625,   1000,    900,    480,   120;
    7776,  12960,  12960,   8640,  3600,   720;
  117649, 201684, 216090, 164640, 88200, 30240, 5040;
  ...

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).
  • I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.

Links

Crossrefs

Column 1: A000169.
Main diagonal: A000142.
T(n, n-1): A062119.
Row sums give A000312.
Cf. A001864, A021010, A063169, A122525, A144084, A243203.

Programs

  • Maple
    T:= (n, k)-> k*n^(n-k)*(n-1)!/(n-k)!:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 22 2012
  • Mathematica
    f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Flatten[Map[f, Drop[Range[0, 10]! CoefficientList[Series[1/(1 - y*t), {x, 0, 10}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 05 2011 *)
  • PARI
    T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ Charles R Greathouse IV, Dec 05 2011

Formula

T(n,k) = k*n^(n-k)*(n-1)!/(n-k)!.
E.g.f. (relative to x): A(x, y)=1/(1-y*B(x)) - 1 = y*x +(2*y+2*y^2)*x^2/2! + (9*y+12*y^2+6*y^3)*x^3/3! + ..., where B(x) is e.g.f. A000169.
From Peter Bala, Sep 30 2011: (Start)
Let F(x,t) = x/(1+t*x)*exp(-x/(1+t*x)) = x*(1 - (1+t)*x + (1+4*t+2*t^2)*x^2/2! - ...). F is essentially the e.g.f. for A144084 (see also A021010). Then the e.g.f. for the present table is t*F(x,t)^(-1), where the compositional inverse is taken with respect to x.
Removing a factor of n from the n-th row entries results in A122525 in row reversed form.
(End)
Sum_{k=2..n} (k-1) * T(n,k) = A001864(n). - Geoffrey Critzer, Aug 19 2013
Sum_{k=1..n} k * T(n,k) = A063169(n). - Alois P. Heinz, Dec 15 2021

A295383 a(n) = (2*n)! * [x^(2*n)] (-x/(1 - x))^n/((1 - x)*n!).

Original entry on oeis.org

1, -4, 72, -2400, 117600, -7620480, 614718720, -59364264960, 6678479808000, -857813628672000, 123868287980236800, -19863969090648883200, 3502679882984419737600, -673592285189311488000000, 140299650258002307072000000, -31464534897861317399347200000
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

Links

Crossrefs

Central terms of triangles A021009 and A021010.
Cf. A144084.

Programs

  • Magma
    R:= RealField(); [Round((-16)^n*Gamma(n+1/2)^2/(Pi(R)*Gamma(n+1) )): n in [0..30]]; // G. C. Greubel, Feb 06 2018
  • Maple
    a := n -> (-16)^n*GAMMA(n+1/2)^2/(Pi*GAMMA(n+1)):
seq(a(n), n=0..15); # Peter Luschny, Nov 21 2017
  • Mathematica
    Table[(2 n)! SeriesCoefficient[(-x/(1 - x))^n /((1 - x) n!), {x, 0, 2 n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[2 EllipticK[-16 x]/Pi, {x, 0, nmax}], x] Range[0, nmax]!
Table[(-16)^n*Gamma[n + 1/2]^2/(Pi*Gamma[n + 1]), {n,0,50}] (* G. C. Greubel, Feb 06 2018 *)
  • PARI
    for(n=0,30, print1(round((-16)^n*gamma(n+1/2)^2/(Pi*gamma(n+1))), ", ")) \\ G. C. Greubel, Feb 06 2018

Formula

E.g.f.: 2*K(-16*x)/Pi, where K() is the complete elliptic integral of the first kind.
a(n) ~ (-1)^n * 16^n * (n-1)! / Pi. - Vaclav Kotesovec, Nov 21 2017
From Peter Luschny, Nov 21 2017: (Start)
a(n) = (-16)^n*Gamma(n+1/2)^2/(Pi*Gamma(n+1)).
a(n) = (-16)^n*binomial(n-1/2,-1/2)*Gamma(n+1/2)/sqrt(Pi).
a(n) ~ (-exp(-1)*n*16)^n/sqrt(n*Pi/2). (End)
a(n) = (-1)^n*binomial(2*n,n)^2*n!. - Alois P. Heinz, Oct 02 2021

A145118 Denominator polynomials for continued fraction generating function for n!.

Original entry on oeis.org

1, 1, 1, -1, 1, -2, 1, -4, 2, 1, -6, 6, 1, -9, 18, -6, 1, -12, 36, -24, 1, -16, 72, -96, 24, 1, -20, 120, -240, 120, 1, -25, 200, -600, 600, -120, 1, -30, 300, -1200, 1800, -720, 1, -36, 450, -2400, 5400, -4320, 720, 1, -42, 630, -4200, 12600, -15120
Offset: 0

Views

Author

Paul Barry, Oct 02 2008

Keywords

Comments

Row sums are A056920. T(n,1) gives quarter squares A002620. T(n,2) appears to coincide with 2*A000241(n+1).

Examples

			Triangle begins:
1;
1;
1,  -1;
1,  -2;
1,  -4,   2;
1,  -6,   6;
1,  -9,  18,    -6;
1, -12,  36,   -24;
1, -16,  72,   -96,   24;
1, -20, 120,  -240,  120;
1, -25, 200,  -600,  600,  -120;
1, -30, 300, -1200, 1800,  -720;
1, -36, 450, -2400, 5400, -4320, 720;

Links

Crossrefs

Cf. A021010, A008297, A066667, A105278.

Programs

  • Maple
    T:= (n, k)-> (-1)^k* binomial(iquo(n+1, 2),k) *binomial(iquo(n, 2), k)*k!:
seq (seq (T(n, k), k=0..iquo(n, 2)), n=0..16);  # Alois P. Heinz, Dec 04 2012

Formula

T(n,k) = (-1)^k C(floor((n+1)/2),k) * C(floor(n/2),k)*k!.

A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 12, 6, 0, 4, 36, 72, 24, 0, 5, 80, 360, 480, 120, 0, 6, 150, 1200, 3600, 3600, 720, 0, 7, 252, 3150, 16800, 37800, 30240, 5040, 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320, 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880
Offset: 0

Views

Author

Peter Luschny, Mar 28 2023

Keywords

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,   2;
  [3] 0, 3,  12,     6;
  [4] 0, 4,  36,    72,     24;
  [5] 0, 5,  80,   360,    480,     120;
  [6] 0, 6, 150,  1200,   3600,    3600,     720;
  [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;
  [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;
  [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;

Crossrefs

Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.
Cf. A144084, A021010.

Programs

  • Maple
    A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
seq(seq(A361893(n,k), k = 0..n), n = 0..9);
# Using the egf.:
egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);

Formula

T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).

A176063 Asymmetrical triangle sequence:t(n,m)=(-1)^m* Binomial[n, m] Pochhammer[ -n, m] - (-1)^n Pochhammer[ -n, n] + (-1)^( n - m)* Binomial[n, -m + n] Pochhammer[ -n, -m + n].

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 88, 120, 88, 1, 1, 505, 680, 680, 505, 1, 1, 3636, 5130, 4080, 5130, 3636, 1, 1, 30289, 48762, 31710, 31710, 48762, 30289, 1, 1, 282304, 525728, 354816, 194880, 354816, 525728, 282304, 1, 1, 2903121, 6171552, 4759776
Offset: 0

Views

Author

Roger L. Bagula, Apr 07 2010

Keywords

Comments

Row sums are:
{1, 7, 43, 297, 2371, 21613, 221523, 2520577, 31515427, 429407661,...}.

Examples

			{1},
{1, 1},
{1, 6, 1},
{1, 21, 21, 1},
{1, 88, 120, 88, 1},
{1, 505, 680, 680, 505, 1},
{1, 3636, 5130, 4080, 5130, 3636, 1},
{1, 30289, 48762, 31710, 31710, 48762, 30289, 1},
{1, 282304, 525728, 354816, 194880, 354816, 525728, 282304, 1},
{1, 2903121, 6171552, 4759776, 1923264, 1923264, 4759776, 6171552, 2903121, 1},
{1, 32659300, 78023250, 69033600, 29181600, 11612160, 29181600, 69033600, 78023250, 32659300, 1}

Crossrefs

Cf. A021010

Programs

  • Mathematica
    L[n_, m_] = (-1)^m* Binomial[n, m] Pochhammer[ -n, m] - (-1)^n Pochhammer[ -n, n] + (-1)^(n - m)* Binomial[n, - m + n] Pochhammer[ -n, -m + n];
Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,m)=(-1)^m* Binomial[n, m] Pochhammer[ -n, m] - (-1)^n Pochhammer[ -n, n] + (-1)^( n - m)* Binomial[n, -m + n] Pochhammer[ -n, -m + n]
Showing 1-7 of 7 results.

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