A182867 -id:A182867 - cp's OEIS frontend

A002455 Central factorial numbers: unsigned 1st subdiagonal of A182867.

0, 1, 20, 784, 52480, 5395456, 791691264, 157294854144, 40683662475264, 13288048674471936, 5349739088314368000, 2603081566154391552000, 1506057980251484454912000, 1021944601582419125993472000

Offset: 0

N. J. A. Sloane

			(arcsin x)^4 = x^4 + 2/3*x^6 + 7/15*x^8 + 328/945*x^10 + ...

B. Berndt, Ramanujan's Notebooks, Part I, page 263.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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=0..50
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
Index entries for sequences related to factorial numbers

Cf. A001819, A001824, A001825, A049033.

List([0..20], n-> 4^(n-1)*(Factorial(n))^2*Sum([1..n], k-> 1/k^2)); # G. C. Greubel, Jul 04 2019

[0] cat [4^(n-1)*(Factorial(n))^2*(&+[1/k^2: k in [1..n]]): n in [1..20]]; // G. C. Greubel, Jul 04 2019

A002455 := proc(n)
    arcsin(x)^4;
    coeftayl(%,x=0,2*n+2)*(2*n+2)!/4! ;
end proc:
seq(A002455(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

nmax = 13; coes = CoefficientList[ Series[ ArcSin[x]^4, {x, 0, 2*nmax + 2}], x]* Range[0, 2*nmax + 2]!/24; a[n_] := coes[[2*n + 3]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 08 2011 *)
Table[4^(n-1)*(n!)^2*HarmonicNumber[n,2], {n,0,20}] (* G. C. Greubel, Jul 04 2019 *)

a(n)=if(n<0,0,(2*n+2)!*polcoeff(asin(x+O(x^(2*n+3)))^4/4!,2*n+2))

a(n)=-(-1)^n*polcoeff(prod(k=0,2*n,x+2*k-2*n),3)

[4^(n-1)*(factorial(n))^2*sum(1/k^2 for k in (1..n)) for n in (0..20)] # G. C. Greubel, Jul 04 2019

(-1)^(n-1)*a(n) is the coefficient of x^3 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: (arcsin x)^4; that is, a_k is the coefficient of x^(2*k+2) in (arcsin x)^4 multiplied by (2*k+2)! and divided by 4! Also a(n) = 2^(2*n-2)*(n!)^2 * Sum_{k=1..n} 1/k^2. - Joe Keane (jgk(AT)jgk.org)
a(n) = 4*(2*n^2 - 2*n + 1)*a(n-1) - 16*(n-1)^4*a(n-2). - Vaclav Kotesovec, Feb 23 2015
a(n) ~ Pi^3 * 2^(2*n-2) * n^(2*n+1) / (3 * exp(2*n)). - Vaclav Kotesovec, Feb 23 2015

More terms from Joe Keane (jgk(AT)jgk.org)

A049033 Central factorial numbers: unsigned 2nd subdiagonal of A182867.

Original entry on oeis.org

1, 56, 4368, 489280, 75851776, 15658639360, 4165906530304, 1390437378293760, 569462999991975936, 280969831084430721024, 164441704270786486861824, 112668650067303149573505024

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

			(arcsin x)^6 = x^6 + x^8 + 13/15*x^10 + 139/189*x^12 + ...

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Cf. A001824, A001825, A002455.

Equals 4^n * A001820(n).

A049033 := proc(n)
    arcsin(x)^6;
    coeftayl(%,x=0,2*n+6)*(2*n+6)!/6! ;
end proc:
seq(A049033(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

E.g.f.: (arcsin x)^6; that is, a_k is the coefficient of x^(2*k+6) in (arcsin x)^6 multiplied by (2*k+6)! and divided by 6!. - Joe Keane (jgk(AT)jgk.org)

(-1)^(n-2)*a(n-2) is the coefficient of x^5 in prod(k=0, 2*n, x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002

A008955 Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 4, 1, 14, 49, 36, 1, 30, 273, 820, 576, 1, 55, 1023, 7645, 21076, 14400, 1, 91, 3003, 44473, 296296, 773136, 518400, 1, 140, 7462, 191620, 2475473, 15291640, 38402064, 25401600, 1, 204, 16422, 669188, 14739153, 173721912, 1017067024, 2483133696, 1625702400

Offset: 0

N. J. A. Sloane

Discussion of Central Factorial Numbers by N. J. A. Sloane, Feb 01 2011: (Start)
Here is Riordan's definition of the central factorial numbers t(n,k) given in Combinatorial Identities, Section 6.5:
For n >= 0, expand the polynomial
x^[n] = x*Product{i=1..n-1} (x+n/2-i) = Sum_{k=0..n} t(n,k)*x^k.
The t(n,k) are not always integers. The cases n even and n odd are best handled separately.
For n=2m, we have:
x^[2m] = Product_{i=0..m-1} (x^2-i^2) = Sum_{k=1..m} t(2m,2k)*x^(2k).
E.g. x^[8] = x^2(x^2-1^2)(x^2-2^2)(x^2-3^2) = x^8-14x^6+49x^4-36x^2,
which corresponds to row 4 of the present triangle.
So the m-th row of the present triangle gives the absolute values of the coefficients in the expansion of Product_{i=0..m-1} (x^2-i^2).
Equivalently, and simpler, the n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first.
For n odd, n=2m+1, we have:
x^[2m+1] = x*Product_{i=0..m-1}(x^2-((2i+1)/2)^2) = Sum_{k=0..m} t(2m+1,2k+1)*x^(2k+1).
E.g. x^[5] = x(x^2-(1/2)^2)(x^2-(3/2)^2) = x^5-10x^3/4+9x/16,
which corresponds to row 2 of the triangle in A008956.
We now rescale to get integers by replacing x by x/2 and multiplying by 2^(2m+1) (getting 1, -10, 9 from the example).
The result is that row m of triangle A008956 gives the coefficients in the expansion of x*Product_{i=0..m} (x^2-(2i+1)^2).
Equivalently, and simpler, the n-th row of A008956 gives the coefficients in the expansion of Product_{i=0..n-1} (x+(2i+1)^2), highest powers first.
Note that the n-th row of A182867 gives the coefficients in the expansion of Product_{i=1..n} (x+(2i)^2), highest powers first.
(End)
Contribution from Johannes W. Meijer, Jun 18 2009: (Start)
We define Beta(n-z,n+z)/Beta(n,n) = Gamma(n-z)*Gamma(n+z)/Gamma(n)^2 = sum(EG2[2m,n]*z^(2m), m = 0..infinity) with Beta(z,w) the Beta function. The EG2[2m,n] coefficients are quite interesting, see A161739. Our definition leads to EG2[2m,1] = 2*eta(2m) and the recurrence relation EG2[2m,n] = EG2[2m,n-1] - EG2[2m-2,n-1]/(n-1)^2 for m = -2, -1, 0, 1, 2, ... and n = 2, 3, ... , with eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. We found for the matrix coefficients EG2[2m,n] = sum((-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2,k=1..n) with the central factorial numbers t1(n,m) as defined above, see also the Maple program.
From the EG2 matrix we arrive at the ZG2 matrix, see A161739 for its odd counterpart, which is defined by ZG2[2m,1] = 2*zeta(2m) and the recurrence relation ZG2[2m,n] = ZG2[2m-2,n-1]/(n*(n-1))-(n-1)*ZG2[2m,n-1]/n for m = -2, -1, 0, 1, 2, ... and n = 2, 3, ... . We found for the ZG2[2m,n] = Sum_{k=1..n} (-1)^(k+1)*t1(n-1,k-1)* 2* zeta(2*m-2*n+2*k)/((n-1)!*(n)!), and we see that the central factorial numbers t1(n,m) once again play a crucial role.
(End)

			Triangle begins:
  1;
  1,   1;
  1,   5,   4;
  1,  14,  49,  36;
  1,  30, 273, 820, 576;
  ...

B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Alois P. Heinz, Rows n = 0..100, flattened (first 51 rows from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
R. H. Boels, Three particle superstring amplitudes with massive legs, arXiv preprint arXiv:1201.2655 [hep-th], 2012.
R. H. Boels and T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt, Central Factorial Numbers: Their main properties and some applications, Numerical Functional Analysis and Optimization, 10 (5&6), 419-488 (1989).
M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
T. L. Curtright and T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arxiv:1408.0767 [math-ph], 2014.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
Toshiki Matsusaka, Applications of Faà di Bruno's formula to partition traces, arXiv:2507.00404 [math.NT], 2025. See p. 5.
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
S. Shadrin, L. Spitz, and D. Zvonkine, On double Hurwitz numbers with completed cycles, J. Lond. Math. Soc., II. Ser. 86, No. 2, 407-432 (2012), Corollary 7.5.

Cf. A036969.
Columns include A000330, A000596, A000597. Right-hand columns include A001044, A001819, A001820, A001821. Row sums are in A101686.
Appears in A160464 (Eta triangle), A160474 (Zeta triangle), A160479 (ZL(n)), A161739 (RSEG2 triangle), A161742, A161743, A002195, A002196, A162440 (EG1 matrix), A162446 (ZG1 matrix) and A163927. - Johannes W. Meijer, Jun 18 2009, Jul 06 2009 and Aug 17 2009
Cf. A234324 (central terms).

T:= function(n,k)
    if k=0 then return 1;
    elif k=n then return (Factorial(n))^2;
    else return n^2*T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..8], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Sep 14 2019

a008955 n k = a008955_tabl !! n !! k
a008955_row n = a008955_tabl !! n
a008955_tabl = [1] : f [1] 1 1 where
   f xs u t = ys : f ys v (t * v) where
     ys = zipWith (+) (xs ++ [t^2]) ([0] ++ map (* u^2) (init xs) ++ [0])
     v = u + 1
-- Reinhard Zumkeller, Dec 24 2013

T:= func< n,k | Factorial(2*(n+1))*(&+[(-1)^j*Binomial(n,k-j)*(&+[2^(m-2*k)*StirlingFirst(2*(n-k+1)+m, 2*(n-k+1))*Binomial(2*(n-k+1)+2*j-1, 2*(n-k+1)+m-1)/Factorial(2*(n-k+1)+m): m in [0..2*j]]): j in [0..k]]) >;
[T(n,k): k in [0..n], n in [0..8]]; // G. C. Greubel, Sep 14 2019

nmax:=7: for n from 0 to nmax do t1(n, 0):=1: t1(n, n):=(n!)^2 end do: for n from 1 to nmax do for k from 1 to n-1 do t1(n, k) := t1(n-1, k-1)*n^2 + t1(n-1, k) end do: end do: seq(seq(t1(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 18 2009, Revised Sep 16 2012
t1 := proc(n,k)
        sum((-1)^j*stirling1(n+1,n+1-k+j)*stirling1(n+1,n+1-k-j),j=-k..k) ;
end proc: # Mircea Merca, Apr 02 2012
# third Maple program:
T:= proc(n, k) option remember; `if`(k=0, 1,
      add(T(j-1, k-1)*j^2, j=1..n))
    end:
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Feb 19 2022

t[n_, 0]=1; t[n_, n_]=(n!)^2; t[n_ , k_ ]:=t[n, k] = n^2*t[n-1, k-1] + t[n-1, k]; Flatten[Table[t[n, k], {n,0,8}, {k,0,n}] ][[1 ;; 42]]
(* Jean-François Alcover, May 30 2011, after recurrence formula *)

T(n,m):=(2*(n+1))!*sum((-1)^k*binomial(n,m-k)*sum((2^(i-2*m)*stirling1(2*(n-m+1)+i,2*(n-m+1))*binomial(2*(n-m+1)+2*k-1,2*(n-m+1)+i-1))/(2*(n-m+1)+i)!,i,0,2*k),k,0,m); /* Vladimir Kruchinin, Oct 05 2013 */

T(n,k)=if(k==0,1, if(k==n, (n!)^2, n^2*T(n-1, k-1) + T(n-1, k)));
for(n=0,8, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 14 2019

# This triangle is (0,0)-based.
def A008955(n, k) :
    if k==0 : return 1
    if k==n : return factorial(n)^2
    return n^2*A008955(n-1, k-1) + A008955(n-1, k)
for n in (0..7) : print([A008955(n, k) for k in (0..n)]) # Peter Luschny, Feb 04 2012

The n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first (see Comments section).
The triangle can be obtained from the recurrence t1(n,k) = n^2*t1(n-1,k-1) + t1(n-1,k) with t1(n,0) = 1 and t1(n,n) = (n!)^2.
t1(n,k) = Sum_{j=-k..k} (-1)^j*s(n+1,n+1-k+j)*s(n+1,n+1-k-j) = Sum_{j=0..2*(n+1-k)} (-1)^(n+1-k+j)*s(n+1,j)*s(n+1,2*(n+1-k)-j), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 02 2012
E.g.f.: cosh(2/sqrt(t)*asin(sqrt(t)*z/2)) = 1 + z^2/2! + (1 + t)*z^4/4! + (1 + 5*t + 4*t^2)*z^6/6! + ... (see Berndt, p.263 and p.306). - Peter Bala, Aug 29 2012
T(n,m) = (2*(n+1))!*Sum_{k=0..m} ((-1)^k*binomial(n,m-k)*Sum_{i=0..2*k} ((2^(i-2*m)*stirling1(2*(n-m+1)+i,2*(n-m+1))*binomial(2*(n-m+1)+2*k-1, 2*(n-m+1)+i-1))/(2*(n-m+1)+i)!)). - Vladimir Kruchinin, Oct 05 2013

There's an error in the last column of Riordan's table (change 46076 to 21076).
More terms from Vladeta Jovovic, Apr 16 2000
Link added and cross-references edited by Johannes W. Meijer, Aug 17 2009
Discussion of Riordan's definition of central factorial numbers added by N. J. A. Sloane, Feb 01 2011

A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.

Original entry on oeis.org

1, 1, -1, 1, -4, 1, -10, 9, 1, -20, 64, 1, -35, 259, -225, 1, -56, 784, -2304, 1, -84, 1974, -12916, 11025, 1, -120, 4368, -52480, 147456, 1, -165, 8778, -172810, 1057221, -893025, 1, -220, 16368, -489280, 5395456, -14745600, 1, -286, 28743, -1234948, 21967231, -128816766, 108056025, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400

Offset: 0

N. J. A. Sloane, Feb 01 2011

These are scaled versions of the central factorial numbers in A008955 and A008956.
Even-indexed rows give A182867, odd-indexed rows give A008956.
A121408 is an unsigned and aerated version of the row reverse of this triangle. - Peter Bala, Aug 29 2012

			Triangle begins:
1
1, -1
1, -4
1, -10, 9
1, -20, 64
1, -35, 259, -225
1, -56, 784, -2304
1, -84, 1974, -12916, 11025
1, -120, 4368, -52480, 147456
1, -165, 8778, -172810, 1057221, -893025
1, -220, 16368, -489280, 5395456, -14745600
...
E.g. for n=5 Q(5) = (x-1^2)*(x-3^2)*(x-5^2) = x^3-35*x^2+259*x-225.

Even-indexed rows give A182867, odd-indexed rows give A008956.
Column 1,4,10,20, ... is A000292. The next two columns give A181888, A184878. The last diagonal is A184877.
Cf. A008955, A008956. A121408.

Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 12 do
t1:=eval(Q(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:

For n even, let Q(n) = Product_{i=1..n/2} (x - (2*i)^2) and for n odd let Q(n) = Product_{i=0..(n-1)/2} (x - (2i+1)^2). n-th row of triangle gives coefficients in expansion of Q(n).

A160563 Table of the number of (n,k)-Riordan complexes, read by rows.

Original entry on oeis.org

1, 1, 1, 9, 10, 1, 225, 259, 35, 1, 11025, 12916, 1974, 84, 1, 893025, 1057221, 172810, 8778, 165, 1, 108056025, 128816766, 21967231, 1234948, 28743, 286, 1, 18261468225, 21878089479, 3841278805, 230673443, 6092515, 77077, 455, 1, 4108830350625, 4940831601000

Offset: 0

Jonathan Vos Post, May 19 2009

From Table 4, right-hand side, of Gelineau and Zeng.

Essentially a row-reversal of A008956. - R. J. Mathar, May 20 2009

			Triangle starts:
  [0]         1;
  [1]         1,          1;
  [2]         9,         10,        1;
  [3]       225,        259,       35,        1;
  [4]     11025,      12916,     1974,       84,     1;
  [5]    893025,    1057221,   172810,     8778,   165,    1;
  [6] 108056025,  128816766, 21967231,  1234948, 28743,  286, 1;
.
For row 3: F(x) := 1/cos(x). Then 225*F(x) + 259*(d/dx)^2(F(x)) + 35*(d/dx)^4(F(x)) + (d/dx)^6(F(x)) = 720*(1/cos(x))^7, where F^(r) denotes the r-th derivative of F(x).

Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 2009.
W. Zhang, Some identities involving the Euler and the central factorial numbers, The Fibonacci Quarterly, Vol. 36, Number 2, May 1998.

Cf. A001819, A008275, A008277, A160562, A091885, A182867.

t := proc(n,k) option remember ; expand(x*mul(x+n/2-i,i=1..n-1)) ; coeftayl(%,x=0,k) ; end:
v := proc(n,k) option remember ; 4^(n-k)*t(2*n+1,2*k+1) ; end:
A160563 := proc(n,k) abs(v(n,k)) ; end: for n from 0 to 10 do for k from 0 to n do printf("%d,",A160563(n,k)) ; od: od: # R. J. Mathar, May 20 2009
# Using a bivariate generating function (albeit generating signed terms):
gf := (t + sqrt(1 + t^2))^x: ser := series(gf, t, 20):
ct := n -> coeff(ser, t, n): T := (n, k) -> n!*coeff(ct(n), x, k):
OddPart := (T, len) -> local n, k;
seq(print(seq(T(n, k), k = 1..n, 2)), n = 1..2*len, 2):
OddPart(T, 6);  # Peter Luschny, Mar 03 2024

t[, 0] = 1; t[n, n_] := t[n, n] = ((2*n - 1)!!)^2; t[n_, k_] := t[n, k] = (2*n - 1)^2*t[n - 1, k - 1] + t[n - 1, k];
T[n_, k_] := t[n, n - k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar's comment *)

a(n,k) = |v(n,k)| where v(n,k) = v(n-1,k-1) - (2n-1)^2*v(n-1,k); eq (4.2).
Let F(x) = 1/cos(x). Then (2*n)!*(1/cos(x))^(2*n+1) = Sum_{k=0..n} T(n,k)*F^(2*k)(x), where F^(r) denotes the r-th derivative of F(x) (Zhang 1998). An example is given below. - Peter Bala, Feb 06 2012
Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=1..n step 2, n=1..len step 2] the 'odd subtriangle' of U. This triangle is the odd subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit with signed terms. See A182867 for the even subtriangle. - Peter Luschny, Mar 03 2024

Extended by R. J. Mathar, May 20 2009

cp's OEIS Frontend

A002455 Central factorial numbers: unsigned 1st subdiagonal of A182867.

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A049033 Central factorial numbers: unsigned 2nd subdiagonal of A182867.

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A008955 Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

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A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)*(x-(n-2)^2)*(x-(n-4)^2)*...*(x-(1 or 2)^2), highest powers first.

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A160563 Table of the number of (n,k)-Riordan complexes, read by rows.

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Formula

Extensions

A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.