A021012 -id:A021012 - cp's OEIS frontend

A136572 Triangle read by rows: row n consists of n zeros followed by n!.

1, 0, 1, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 0, 0, 5040, 0, 0, 0, 0, 0, 0, 0, 0, 40320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800

Offset: 0

Gary W. Adamson, Jan 07 2008

A136572 * A007318 = A021012(unsigned). A007318 * A136572 = A008279.

			First few rows of the triangle:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 0, 6;
  0, 0, 0, 0, 24;
  0, 0, 0, 0,  0, 120;
  ...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Cf. A000142, A021012, A008279.

a136572 n k = a136572_tabl !! n !! k
a136572_row n = a136572_tabl !! n
a136572_tabl = map fst $ iterate f ([1], 1) where
   f (row, i) = (0 : map (* i) row, i + 1)
-- Reinhard Zumkeller, Nov 18 2012

Table[PadLeft[{n!},n+1,0],{n,0,20}]//Flatten (* Harvey P. Dale, Oct 22 2016 *)

Triangle, n zeros followed by n! T(n,k): n! * 0^(n-k), 0 <= k <= n.

As an infinite lower triangular matrix, A000142 (1, 1, 2, 6, 24, 120, ...) in the main diagonal and the rest zeros.

A001805 a(n) = n! * binomial(n,3).

Original entry on oeis.org

6, 96, 1200, 14400, 176400, 2257920, 30481920, 435456000, 6586272000, 105380352000, 1780927948800, 31732897996800, 594991837440000, 11716762337280000, 241867451105280000, 5224336943874048000, 117874102296158208000, 2773508289321369600000

Offset: 3

N. J. A. Sloane

nonn

Coefficients of Laguerre polynomials.

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.
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 = 3..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Laguerre polynomials

Essentially a column of triangle A021012.

[Factorial(n)*Binomial(n,3): n in [3..30]]; // G. C. Greubel, May 17 2018

G(x):=x^3/(1-x)^4: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=3..18); # Zerinvary Lajos, Apr 01 2009

Table[n! Binomial[n,3], {n,3,30}] (* Harvey P. Dale, Feb 23 2011 *)

for(n=3, 30, print1(n!*binomial(n,3), ", ")) \\ G. C. Greubel, May 17 2018

E.g.f.: x^3/(1-x)^4. - Geoffrey Critzer, Aug 19 2012

More terms from Ralf Stephan, Jan 09 2004

A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320

Offset: 0

Philippe Deléham, Oct 28 2011

Unsigned version of A021012.

Equal to A136572*A007318.

			Triangle begins:
    1;
    1,   1;
    2,   4,    2;
    6,  18,   18,    6;
   24,  96,  144,   96,  24;
  120, 600, 1200, 1200, 600, 120;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..495
P. Bala, Deformations of the Hadamard product of power series
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Cf. A007318, A008619, A021012, A084938, A136572.

Columns include: A000142, A001563, A001804, A001805, A001806, A001807.

/* As triangle */ [[Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 28 2015

Table[n!*Binomial[n, j], {n, 0, 30}, {j, 0, n}] (* G. C. Greubel, Sep 27 2015 *)

factorial(n)*binomial(n,k) # Danny Rorabaugh, Sep 27 2015

T(n,k) is given by (1,1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,1,2,2,3,3,4,4,5,5,6,6, ...) where DELTA is the operator defined in A084938.
Sum_{k>=0} T(m,k)*T(n,k) = (m+n)!.
T(2n,n) = A122747(n).
Sum_{k>=0} T(n,k)^2 = A010050(n) = (2n)!.
Sum_{k>=0} T(n,k)*x^k = A000007(n), A000142(n), A000165(n), A032031(n), A047053(n), A052562(n), A047058(n), A051188(n), A051189(n), A051232(n), A051262(n), A196258(n), A145448(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
The row polynomials have the form (x + 1) o (x + 2) o ... o (x + n), where o denotes the black diamond multiplication operator of Dukes and White. See example E10 in the Bala link. - Peter Bala, Jan 18 2018

Name exchanged with a formula by Peter Luschny, Feb 01 2015

A001806 a(n) = n! * binomial(n,4).

Original entry on oeis.org

24, 600, 10800, 176400, 2822400, 45722880, 762048000, 13172544000, 237105792000, 4452319872000, 87265469491200, 1784975512320000, 38079477596160000, 846536078868480000, 19591263539527680000, 471496409184632832000, 11787410229615820800000

Offset: 4

N. J. A. Sloane

nonn

Coefficients of Laguerre polynomials.

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.
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 = 4..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Laguerre polynomials

A column of triangle A021012.

[Factorial(n)*Binomial(n,4): n in [4..30]]; // G. C. Greubel, May 17 2018

G(x):=x^4/(1-x)^5: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=4..18); # Zerinvary Lajos, Apr 01 2009

Table[n! Binomial[n, 4], {n, 4, 20}] (* T. D. Noe, Aug 10 2012 *)

for(n=4,30, print1(n!*binomial(n,4), ", ")) \\ G. C. Greubel, May 17 2018

E.g.f.: x^4/(1-x)^5. - Geoffrey Critzer, Aug 19 2012

More terms from Ralf Stephan, Jan 09 2004

A001807 a(n) = n! * binomial(n,5).

Original entry on oeis.org

120, 4320, 105840, 2257920, 45722880, 914457600, 18441561600, 379369267200, 8014175769600, 174530938982400, 3926946127104000, 91390746230784000, 2200993805058048000, 54855537910677504000, 1414489227553898496000, 37719712734770626560000

Offset: 5

N. J. A. Sloane

nonn

Coefficients of Laguerre polynomials.

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.
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 = 5..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Laguerre polynomials

Essentially a column of triangle A021012.

[Factorial(n)*Binomial(n,5): n in [5..35]]; // G. C. Greubel, May 17 2018

G(x):=x^5/(1-x)^6: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=5..18); # Zerinvary Lajos, Apr 01 2009

Table[n! Binomial[n, 5], {n, 5, 20}] (* T. D. Noe, Aug 10 2012 *)

for(n=5, 35, print1(n!*binomial(n,5), ", ")) \\ G. C. Greubel, May 17 2018

[binomial(n,5)*factorial (n) for n in range(5, 19)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: x^5/(1-x)^6. - Geoffrey Critzer, Aug 19 2012

More terms from Ralf Stephan, Jan 09 2004

A174690 Triangle T(n, k) = n!*binomial(n, k) - n! + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 13, 13, 1, 1, 73, 121, 73, 1, 1, 481, 1081, 1081, 481, 1, 1, 3601, 10081, 13681, 10081, 3601, 1, 1, 30241, 100801, 171361, 171361, 100801, 30241, 1, 1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1, 1, 2903041, 12700801, 30119041, 45360001, 45360001, 30119041, 12700801, 2903041, 1

Offset: 0

Roger L. Bagula, Mar 27 2010

			Triangle begins as:
  1;
  1,      1;
  1,      3,       1;
  1,     13,      13,       1;
  1,     73,     121,      73,       1;
  1,    481,    1081,    1081,     481,       1;
  1,   3601,   10081,   13681,   10081,    3601,       1;
  1,  30241,  100801,  171361,  171361,  100801,   30241,      1;
  1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A000165, A021012, A196347.

[Factorial(n)*(Binomial(n,k) -1) + 1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

T[n_, k_]:= n!*Binomial[n, k] - n! + 1;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

flatten([[factorial(n)*(binomial(n,k) -1) + 1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021

T(n, k) = n!*binomial(n, k) - n! + 1.
From G. C. Greubel, Feb 09 2021: (Start)
T(n, k) = A196347(n, k) - n! + 1 = (-1)^k * A021012(n, k) - n! + 1.
Sum_{k=0..n} T(n, k) = 2^n * n! - (n+1)! + (n+1) = A000165(n) - (n+1)! + (n+1). (End)

Edited by G. C. Greubel, Feb 09 2021

A253666 Triangle read by rows, T(n,k) = C(n,k)*n!/(floor(n/2)!)^2, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 6, 24, 36, 24, 6, 30, 150, 300, 300, 150, 30, 20, 120, 300, 400, 300, 120, 20, 140, 980, 2940, 4900, 4900, 2940, 980, 140, 70, 560, 1960, 3920, 4900, 3920, 1960, 560, 70, 630, 5670, 22680, 52920, 79380, 79380, 52920, 22680, 5670, 630

Offset: 0

Peter Luschny, Feb 01 2015

			Triangle begins:
.   1;
.   1,    1;
.   2,    4,     2;
.   6,   18,    18,     6;
.   6,   24,    36,    24,     6;
.  30,  150,   300,   300,   150,    30;
.  20,  120,   300,   400,   300,   120,    20;
. 140,  980,  2940,  4900,  4900,  2940,   980,   140;
.  70,  560,  1960,  3920,  4900,  3920,  1960,   560,   70;
. 630, 5670, 22680, 52920, 79380, 79380, 52920, 22680, 5670, 630; etc.

Cf. A021012, A196347, A002894.

Row sums are A253665.

[Binomial(n,k)*Factorial(n)/Factorial(Floor(n/2))^2: k in [0..n], n in [0..10]]; // Bruno Berselli, Feb 02 2015

T := (n,k) -> n!*binomial(n,k)/(iquo(n,2)!)^2:
seq(print(seq(T(n,k), k=0..n)), n=0..9);

T(n,k) = C(n,k)*A056040(k).

T(2*n,n) = C(2*n,n)^2.

A360283 a(n) = lcm({n! * binomial(n, k) for k = 0..n}).

Original entry on oeis.org

1, 1, 4, 18, 288, 1200, 43200, 529200, 11289600, 91445760, 9144576000, 92207808000, 13277924352000, 160283515392000, 2094371267788800, 58904191906560000, 15079473128079360000, 242109318556385280000, 78443419212268830720000, 1415903716781452394496000

Offset: 0

Peter Luschny, Feb 14 2023

nonn

Cf. A002397, A003418 (LCM), A005722, A021012, A196347.

a := n -> ilcm(seq(n!*binomial(n, k), k=0..n)):
seq(a(n), n = 0..19);

from math import factorial, lcm
def A360283(n): return factorial(n)*lcm(*(i for i in range(1,n+2)))//(n+1) # Chai Wah Wu, Feb 15 2023

a(n) = n! * lcm({k for k = 1..n+1}) / (n+1) = n! * LCM(n + 1) / (n + 1).

a(n) / a(n-1) = n^2 if and only if n + 1 is prime, for n >= 1.

cp's OEIS Frontend

A136572 Triangle read by rows: row n consists of n zeros followed by n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A001805 a(n) = n! * binomial(n,3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A001806 a(n) = n! * binomial(n,4).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A001807 a(n) = n! * binomial(n,5).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A174690 Triangle T(n, k) = n!*binomial(n, k) - n! + 1, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs