A000407 -id:A000407 - cp's OEIS frontend

A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 12, 0, 12, 0, 1, 0, 60, 0, 20, 0, 1, 120, 0, 180, 0, 30, 0, 1, 0, 840, 0, 420, 0, 42, 0, 1, 1680, 0, 3360, 0, 840, 0, 56, 0, 1, 0, 15120, 0, 10080, 0, 1512, 0, 72, 0, 1, 30240, 0, 75600, 0, 25200, 0, 2520, 0, 90, 0, 1

Offset: 0

Christian G. Bower, Jan 03 2002

x^n = (1/2^n) * Sum_{k=0..n} a(n,k)*H_k(x).

These polynomials, H_n(x), are an Appell sequence, whose umbral compositional inverse sequence HI_n(x) consists of the same polynomials signed with the e.g.f. e^{-t^2} e^{xt}. Consequently, under umbral composition H_n(HI.(x)) = x^n = HI_n(H.(x)). Other differently scaled families of Hermite polynomials are A066325, A099174, and A060821. See Griffin et al. for a relation to the Catalan numbers and matrix integration. - Tom Copeland, Dec 27 2020

			Triangle begins with:
    1;
    0,   1;
    2,   0,   1;
    0,   6,   0,   1;
   12,   0,  12,   0,   1;
    0,  60,   0,  20,   0,   1;
  120,   0, 180,   0,  30,   0,   1;

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801. (Table 22.12)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
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].
M. Griffin, K. Ono, L. Rolen, and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321 [math.NT], 2019.
Index entries for sequences related to Hermite polynomials

Row sums give A047974. Columns 0-2: A001813, A000407, A001814. Cf. A048854, A060821.

Cf. A060821, A066325, and A099174.

[[Round(Factorial(n)*(1+(-1)^(n+k))/(2*Factorial(k)*Gamma((n-k+2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jun 09 2018

T := proc(n, k) (n - k)/2; `if`(%::integer, (n!/k!)/%!, 0) end:
for n from 0 to 11 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jan 05 2021

Table[n!*(1+(-1)^(n+k))/(2*k!*Gamma[(n-k+2)/2]), {n,0,20}, {k,0,n}]// Flatten (* G. C. Greubel, Jun 09 2018 *)

T(n, k) = round(n!*(1+(-1)^(n+k))/(2*k! *gamma((n-k+2)/2)))
for(n=0,20, for(k=0,n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jun 09 2018

{T(n,k) = if(k<0 || nMichael Somos, Jan 15 2020 */

E.g.f. (rel to x): A(x, y) = exp(x*y + x^2).
Sum_{ k>=0 } 2^k*k!*T(m, k)*T(n, k) = T(m+n, 0) = |A067994(m+n)|. - Philippe Deléham, Jul 02 2005
T(n, k) = 0 if n-k is odd; T(n, k) = n!/(k!*((n-k)/2)!) if n-k is even. - Philippe Deléham, Jul 02 2005
T(n, k) = n!/(k!*2^((n-k)/2)*((n-k)/2)!)*2^((n+k)/2)*(1+(-1)^(n+k))/2^(k+1).
T(n, k) = A001498((n+k)/2, (n-k)/2)2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1). - Paul Barry, Aug 28 2005
Exponential Riordan array (e^(x^2),x). - Paul Barry, Sep 12 2006
G.f.: 1/(1-x*y-2*x^2/(1-x*y-4*x^2/(1-x*y-6*x^2/(1-x*y-8*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
The n-th row entries may be obtained from D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. - Peter Bala, Dec 07 2011
As noted in the comments this is an Appell sequence of polynomials, so the lowering and raising operators defined by L H_n(x) = n H_{n-1}(x) and R H_{n}(x) = H_{n+1}(x) are L = D_x, the derivative, and R = D_t log[e^{t^2} e^{xt}] |{t = D_x} = x + 2 D_x, and the polynomials may also be generated by e^{-D^2} x^n = H_n(x). - _Tom Copeland, Dec 27 2020

A087299 Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).

Original entry on oeis.org

1, 1, 4, 6, 32, 60, 384, 840, 6144, 15120, 122880, 332640, 2949120, 8648640, 82575360, 259459200, 2642411520, 8821612800, 95126814720, 335221286400, 3805072588800, 14079294028800, 167423193907200, 647647525324800

Offset: 0

Eric W. Weisstein, Aug 31 2003

			The volume of sphere (3-ball) is 4/3*Pi*r^3 and circumscribing 3-cube is 2^3*r^3 so ratio is Pi/6 and a(3)=6.
G.f. =  1 + x + 4*x^2 + 6*x^3 + 32*x^4 + 60*x^5 + 384*x^6 + 840*x^7 + ...

N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292

G. C. Greubel, Table of n, a(n) for n = 0..725
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Ball

Cf. A000407, A047053, A072345, A072346.

a[ n_] := If[ n < 0, 0, With[ {m = n + 1}, m! SeriesCoefficient[ Exp[x^2] (1 + Sqrt[Pi] Erf[x]), {x, 0, m}] / 2]]; (* Michael Somos, Jan 24 2014 *)
Table[2^n*Gamma[n/2 + 1]*Pi^Floor[n/2]/Pi^(n/2), {n,0,50}] (* G. C. Greubel, Jan 28 2017 *)

{a(n) = my(A); if( n<0, 0, n++; A = exp(x^2 + x * O(x^n)); n! * polcoeff( A * (1 + 2 * intformal( 1/A )), n) / 2)}; /* Michael Somos, May 25 2004 */

{a(n) = if( n<2, n>-1, 2*n * a(n-2))}; /* Michael Somos, Jan 24 2014 */

{a(n) = if( n<0, 0, if( n%2, n! / (n\2)!, 2^n * (n\2)!))}; /* Michael Somos, Jan 03 2015 */

a(n) = 2^n*gamma(n/2+1)*Pi^floor(n/2)/Pi^(n/2), n >= 0. - Wolfdieter Lang, Jul 17 2013
0 = a(n)*( 2*a(n+1) - a(n+3) ) + a(n+1)*a(n+2) if n>=0. - Michael Somos, Jan 24 2014
a(n) = 2*n * a(n-2) if n>=2. - Michael Somos, Jan 24 2014
a(2*n) = A047053(n). a(2*n + 1) = A000407(n). - Michael Somos, Jan 03 2015

A304334 T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 6, 0, 1, 30, 60, 0, 1, 126, 840, 840, 0, 1, 510, 8820, 25200, 15120, 0, 1, 2046, 84480, 526680, 831600, 332640, 0, 1, 8190, 780780, 9609600, 30270240, 30270240, 8648640, 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200

Offset: 0

Peter Luschny, May 11 2018

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 1,     6
[3] 0, 1,    30,      60
[4] 0, 1,   126,     840,       840
[5] 0, 1,   510,    8820,     25200,     15120
[6] 0, 1,  2046,   84480,    526680,    831600,     332640
[7] 0, 1,  8190,  780780,   9609600,  30270240,   30270240,    8648640
[8] 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200

Row sums are bisection of A081562, T(n,n) ~ A000407, T(n,n-1) ~ A048854(n,2), T(n,2) ~ A002446.

Cf. A304330, A304336.

A304334 := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/k!:
for n from 0 to 8 do seq(A304334(n, k), k=0..n) od;

T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n))/k!;
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 11 2018

T(n, k) = A304330(n, k) / k!.

A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -5, 6, 1, -12, 47, -60, 1, -22, 179, -638, 840, 1, -35, 485, -3325, 11274, -15120, 1, -51, 1075, -11985, 74524, -245004, 332640, 1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640, 1, -92, 3682, -83720, 1182769

Offset: 0

N. J. A. Sloane, Apr 18 2000

			Triangle begins:
  1;
  1,  -1;
  1,  -5,    6;
  1, -12,   47,    -60;
  1, -22,  179,   -638,    840;
  1, -35,  485,  -3325,  11274,   -15120;
  1, -51, 1075, -11985,  74524,  -245004,  332640;
  1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640;
  ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened).
Takao Komatsu, Some explicit values of a q-multiple zeta function at roots of unity, arXiv:2505.09357 [math.NT], 2025. See p. 10.

Cf. A000407, A054649, A054651, A054654, A008276.

a054655_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (n-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010

row[n_] := Table[ Coefficient[(-1)^n*Pochhammer[n - x, n], x, n - k], {k, 0, n}]; A054655 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Apr 06 2012, after Maple *)

T(n,k)=polcoef(n!*binomial(x-n,n), n-k);

n!*binomial(x-n, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
From Robert Israel, Jul 12 2016: (Start)
G.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k  = hypergeom([1, -1/(2*y), (1/2)*(-1+y)/y], [-1/y], -4*x*y).
E.g.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! = (1+4*x*y)^(-1/2)*((1+sqrt(1+4*x*y))/2)^(1+1/y). (End)

A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 8, 60, 1, 1, 10, 105, 840, 1, 1, 12, 162, 1920, 15120, 1, 1, 14, 231, 3640, 45045, 332640, 1, 1, 16, 312, 6144, 104720, 1290240, 8648640, 1, 1, 18, 405, 9576, 208845, 3674160, 43648605, 259459200, 1, 1, 20, 510, 14080, 375000, 8648640, 152152000, 1703116800, 8821612800

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

			Square array begins:
      1,      1,       1,       1,       1,       1,  ...
      1,      1,       1,       1,       1,       1,  ...
      6,      8,      10,      12,      14,      16,  ...
     60,    105,     162,     231,     312,     405,  ...
    840,   1920,    3640,    6144,    9576,   14080,  ...
  15120,  45045,  104720,  208845,  375000,  623645,  ...
=========================================================
A(1,1) = 1;
A(2,1) = 2*3 = 6;
A(3,1) = 3*4*5 = 60;
A(4,1) = 4*5*6*7 = 840;
A(5,1) = 5*6*7*8*9 = 15120, etc.
...
A(1,2) = 1;
A(2,2) = 2*4 = 8;
A(3,2) = 3*5*7 = 105;
A(4,2) = 4*6*8*10 = 1920;
A(5,2) = 5*7*9*11*13 = 45045, etc.
...
A(1,3) = 1;
A(2,3) = 2*5 = 10;
A(3,3) = 3*6*9 = 162;
A(4,3) = 4*7*10*13 = 3640;
A(5,3) = 5*8*11*14*17 = 104720, etc.
...

Index entries for sequences related to factorial numbers

Columns k=1..5 give A000407, A113551, A303486, A303487, A303488.
Main diagonal gives A061711.
Cf. A008279, A131182, A256268, A265609.

Table[Function[k, n! SeriesCoefficient[1/(1 - k x)^(n/k), {x, 0, n}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, Product[k i + n, {i, 0, n - 1}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, k^n Pochhammer[n/k, n]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten

A(n,k) = Product_{j=0..n-1} (k*j + n).

A384164 a(n) = Product_{k=0..n-1} (3*n+k).

Original entry on oeis.org

1, 3, 42, 990, 32760, 1395360, 72681840, 4475671200, 318073392000, 25622035084800, 2306992893004800, 229601607198163200, 25028504609870361600, 2965681982933429760000, 379534960108578193920000, 52170410224819317150720000, 7666009844358186506465280000, 1199151678674216896627654656000

Offset: 0

Seiichi Manyama, May 21 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A384164, A384165.
Cf. A000407, A352601.
Cf. A061924, A124320.

[1] cat  [&*[(3*n + k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025

a[n_]:=Product[(3*n+k),{k,0,n-1}]; Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)

PARI
```
a(n) = prod(k=0, n-1, 3*n+k);
```

from sympy import rf
def A384164(n): return rf(3*n,n) # Chai Wah Wu, May 21 2025

def a(n): return rising_factorial(3*n, n)

a(n) = RisingFactorial(3*n,n) = A124320(3*n,n) = n! * binomial(4*n-1,n).
a(n) = n! * [x^n] 1/(1 - x)^(3*n).
a(n) = (3/4) * A061924(n) for n > 0.

A002691 a(n) = (n+2) * (2n+1) * (2n-1)! / (n-1)!.

Original entry on oeis.org

1, 9, 120, 2100, 45360, 1164240, 34594560, 1167566400, 44108064000, 1843717075200, 84475764172800, 4209708914611200, 226676633863680000, 13114862387827200000, 811372819726909440000, 53449184499510159360000, 3735154775612827607040000

Offset: 0

N. J. A. Sloane

Coefficients of orthogonal polynomials.

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).

Alois P. Heinz, Table of n, a(n) for n = 0..150
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177.
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177. [Annotated scanned copy]

Cf. A002690.

with(combstruct): a:=n-> add((count(Permutation(n*2+1), size=n+1)), j=0..n+1)/2: seq(a(n), n=0..16); # Zerinvary Lajos, May 03 2007

Join[{1},Table[(n+2)(2n+1)(2n-1)!/(n-1)!,{n,15}]] (* Harvey P. Dale, Jun 09 2011 *)

PARI
```
a(n)=(n+2)*(2*n+1)*(2*n-1)!/(n-1)!
```

E.g.f.: (1-x)/(1-4*x)^(5/2).

Conjecture: a(n) +4*(-n-1)*a(n-1) +4*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

Edited by Ralf Stephan, Mar 21 2004

A051620 a(n) = (4n+8)(!^4)/8(!^4), related to A034177(n+1) ((4n+4)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 12, 192, 3840, 92160, 2580480, 82575360, 2972712960, 118908518400, 5231974809600, 251134790860800, 13059009124761600, 731304510986649600, 43878270659198976000, 2808209322188734464000, 190958233908833943552000

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(12/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

G(x):=(1-4*x)^(n-4): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=0..15); # Zerinvary Lajos, Apr 04 2009

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn=20},CoefficientList[Series[1/(1-4*x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 10 2017 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(12/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+8)(!^4))/8(!^4) = A034177(n+2)/8.
E.g.f.: 1/(1-4*x)^3.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+6)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013

A002690 a(n) = (n+1) * (2*n)! / n!.

Original entry on oeis.org

1, 4, 36, 480, 8400, 181440, 4656960, 138378240, 4670265600, 176432256000, 7374868300800, 337903056691200, 16838835658444800, 906706535454720000, 52459449551308800000, 3245491278907637760000, 213796737998040637440000, 14940619102451310428160000, 1103945744792235714969600000

Offset: 0

N. J. A. Sloane

Coefficients of orthogonal polynomials.
E.g.f. for series with alternating signs: x/(1+4*x)^(1/2).
Central terms of triangle A245334. - Reinhard Zumkeller, Aug 30 2014

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177.
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177. [Annotated scanned copy]

a(n) = (n+1) * A001813(n) = 2^n * A001193(n+1).
Cf. A002691, A000407.
Cf. A245334.

a002690 n = a245334 (2 * n) n  -- Reinhard Zumkeller, Aug 30 2014

[(n+1) * Factorial(2*n) /Factorial(n): n in [0..20]]; // Vincenzo Librandi, Sep 05 2011

with(combstruct):bin := {B=Union(Z,Prod(B,B))}:
seq (count([B,bin,labeled],size=n+1)*(n+1), n=0..17); # Zerinvary Lajos, Dec 05 2007
A002690 := n -> 2^n*n!*JacobiP(n, -1/2, -n+1, 3):
seq(simplify(A002690(n)), n = 0..18);  # Peter Luschny, Jan 22 2025

Table[((n+1)(2n)!)/n!,{n,0,20}] (* Harvey P. Dale, Sep 04 2011 *)

PARI
```
a(n)=(n+1)*(2*n)!/n!
```

E.g.f.: (1-2*x)/(1-4*x)^(3/2).

a(n) = 2^n*n!*JacobiP(n, -1/2, -n+1, 3). - Peter Luschny, Jan 22 2025

Edited by Ralf Stephan, Mar 21 2004

A051619 a(n) = (4n+7)(!^4)/7(!^4), related to A034176(n+1) ((4n+3)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 11, 165, 3135, 72105, 1946835, 60351885, 2112315975, 82380323025, 3542353890075, 166490632833525, 8491022274509775, 467006225098037625, 27553367280784219875, 1735862138689405852125, 116302763292190192092375

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(11/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 10, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(11/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(11/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+7)(!^4))/7(!^4) = A034176(n+2)/7.

E.g.f.: 1/(1-4*x)^(11/4).

cp's OEIS Frontend

A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

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A087299 Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).

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A304334 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

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A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.

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A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

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A384164 a(n) = Product_{k=0..n-1} (3*n+k).

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A002691 a(n) = (n+2) * (2n+1) * (2n-1)! / (n-1)!.

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A304334 T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

A051620 a(n) = (4n+8)(!^4)/8(!^4), related to A034177(n+1) ((4n+4)(!^4) quartic, or 4-factorials).

A051619 a(n) = (4n+7)(!^4)/7(!^4), related to A034176(n+1) ((4n+3)(!^4) quartic, or 4-factorials).