A001825 -id:A001825 - cp's OEIS frontend

A008956 Triangle of central factorial numbers |4^k t(2n+1,2n+1-2k)| read by rows (n>=0, k=0..n).

1, 1, 1, 1, 10, 9, 1, 35, 259, 225, 1, 84, 1974, 12916, 11025, 1, 165, 8778, 172810, 1057221, 893025, 1, 286, 28743, 1234948, 21967231, 128816766, 108056025, 1, 455, 77077, 6092515, 230673443, 3841278805, 21878089479, 18261468225, 1, 680

Offset: 0

N. J. A. Sloane

The n-th row gives the coefficients in the expansion of Product_{i=0..n-1} (x+(2i+1)^2), highest powers first (see the discussion of central factorial numbers in A008955). - N. J. A. Sloane, Feb 01 2011
Descending row polynomials in x^2 evaluated at k generate odd coefficients of e.g.f. sin(arcsin(kt)/k): 1, x^2 - 1, 9x^4 - 10x^2 + 1, 225x^6 - 259x^4 + 34x^2 - 1, ... - Ralf Stephan, Jan 16 2005
From Johannes W. Meijer, Jun 18 2009: (Start)
We define (Pi/2)*Beta(n-1/2-z/2,n-1/2+z/2)/Beta(n-1/2,n-1/2) = (Pi/2)*Gamma(n-1/2-z/2)* Gamma(n-1/2+z/2)/Gamma(n-1/2)^2 = sum(BG2[2m,n]*z^(2m), m = 0..infinity) with Beta(z,w) the Beta function. Our definition leads to BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .We found for the BG2[2*m,n] = sum((-1)^(k+n)*t2(n-1,k-1)* 2*beta(2*m-2*n+2*k+1),k=1..n)/((2*n-3)!!)^2 with the central factorial numbers t2(n,m) as defined above; see also the Maple program.
From the BG2 matrix and the closely related EG2 and ZG2 matrices, see A008955, we arrive at the LG2 matrix which is defined by LG2[2m-1,1] = 2*lambda(2*m) and the recurrence relation LG2[2*m-1,n] = LG2[2*m-3,n-1]/((2*n-3)*(2*n-1)) - (2*n-3)*LG2[2*m-1,n-1]/(2*n-1) for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with lambda(m) = (1-2^(-m))*zeta(m) with zeta(m) the Riemann zeta function. We found for the matrix coefficients LG2[2m-1,n] = sum((-1)^(k+1)* t2(n-1,k-1)*2*lambda(2*m-2*n+2*k)/((2*n-1)!!*(2*n-3)!!), k=1..n) and we see that the central factorial numbers t2(n,m) once again play a crucial role.
(End)

			Triangle begins:
[1]
[1, 1]
[1, 10, 9]
[1, 35, 259, 225]
[1, 84, 1974, 12916, 11025]
[1, 165, 8778, 172810, 1057221, 893025]
[1, 286, 28743, 1234948, 21967231, 128816766, 108056025]
[1, 455, 77077, 6092515, 230673443, 3841278805, 21878089479, 18261468225]
...

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). [From Johannes W. Meijer, Jun 18 2009]
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Reinhard Zumkeller, 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, Chapter 23, pp. 811-812. [From _Johannes W. Meijer_, Jun 18 2009]
R. H. Boels, T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
T. L. Curtright, D. B. Fairlie, C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
T. L. Curtright, T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
M. Eastwood and H. Goldschmidt, Zero-energy fields on complex projective space, arXiv preprint arXiv:1108.1602 [math.DG], 2011.
M. Eastwood, The X-ray transform on projective space. - From _N. J. A. Sloane_, Oct 22 2012

Cf. A008958.
Columns include A000447, A001823. Right-hand columns include A001818, A001824, A001825. Cf. A008955.
Appears in A160480 (Beta triangle), A160487 (Lambda triangle), A160479 (ZL(n) sequence), A161736, A002197 and A002198. - Johannes W. Meijer, Jun 18 2009
Cf. A162443 (BG1 matrix) and A162448 (LG1 matrix). - Johannes W. Meijer, Jul 06 2009
Cf. A001147.

a008956 n k = a008956_tabl !! n !! k
a008956_row n = a008956_tabl !! n
a008956_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 + 2
-- Reinhard Zumkeller, Dec 24 2013

f:=n->mul(x+(2*i+1)^2,i=0..n-1);
for n from 0 to 12 do
t1:=eval(f(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od: # N. J. A. Sloane, Feb 01 2011
A008956 := proc(n,k) local i ; mul( x+2*i-2*n-1,i=1..2*n) ; expand(%) ; coeftayl(%,x=0,2*(n-k)) ; abs(%) ; end: for n from 0 to 10 do for k from 0 to n do printf("%a,",A008956(n,k)) ; od: od: # R. J. Mathar, May 29 2009
nmax:=7: for n from 0 to nmax do t2(n, 0):=1: t2(n, n):=(doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do t2(n, k) := (2*n-1)^2*t2(n-1, k-1)+t2(n-1, k) od: od: seq(seq(t2(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 18 2009, Revised Sep 16 2012

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]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014, after Johannes W. Meijer *)

{T(n, k) = if( n<=0, k==0, (-1)^k * polcoeff( numerator( 2^(2*n -1) / sum(j=0, 2*n - 1, binomial( 2*n - 1, j) / (x + 2*n - 1 - 2*j))), 2*n - 2*k))}; /* Michael Somos, Feb 24 2003 */

Conjecture row sums: Sum_{k=0..n} T(n,k) = |A101927(n+1)|. - R. J. Mathar, May 29 2009

May be generated by the recurrence t2(n,k) = (2*n-1)^2*t2(n-1,k-1)+t2(n-1,k) with t2(n,0) = 1 and t2(n,n)=((2*n-1)!!)^2. - Johannes W. Meijer, Jun 18 2009

More terms from Vladeta Jovovic, Apr 16 2000

Edited by N. J. A. Sloane, Feb 01 2011

A001824 Central factorial numbers: 1st subdiagonal of A008956.

Original entry on oeis.org

1, 10, 259, 12916, 1057221, 128816766, 21878089479, 4940831601000, 1432009163039625, 518142759828635250, 228929627246078500875, 121292816354463333793500, 75908014254880833434338125, 55399444912646408707007883750, 46636497509226736668824289999375

Offset: 0

N. J. A. Sloane

			(arcsin x)^3 = x^3 + 1/2*x^5 + 37/120*x^7 + 3229/15120*x^9 + ...

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 223, Problem 2.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..224 (terms 0..50 from T. D. Noe)
Index entries for sequences related to factorial numbers

Cf. A002455, A001825, A049033.

Right-hand column 2 in triangle A008956.

a[n_] = (2n+1)!!^2 (Pi^2 - 2 PolyGamma[1, n+3/2])/8; a /@ Range[0, 12] // Simplify (* Jean-François Alcover, Apr 22 2011, after Joe Keane *)
With[{nn=30},Take[(CoefficientList[Series[ArcSin[x]^3,{x,0,nn}], x] Range[0,nn-1]!)/6,{4,-1,2}]] (* Harvey P. Dale, Feb 05 2012 *)

E.g.f.: (arcsin x)^3; that is, a_k is the coefficient of x^(2*k+3) in (arcsin x)^3 multiplied by (2*k+3)! and divided by 6. - Joe Keane (jgk(AT)jgk.org)
a(n) = ((2*n+1)!!)^2 * Sum_{k=0..n} (2*k+1)^(-2).
a(n) ~ Pi^2*n^2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
(-1)^(n-1)*a(n-1) is the coefficient of x^2 in Product_{k=1..2*n} (x + 2*k - 2*n - 1). - Benoit Cloitre and Michael Somos, Nov 22 2002
a(n) = det(V(i+2,j+1), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices (A008958). - Mircea Merca, Apr 06 2013
Recurrence: a(n) = 2*(4*n^2+1)*a(n-1) - (2*n-1)^4*a(n-2). - Vladimir Reshetnikov, Oct 13 2016
Limit_{n->infinity} a(n)/((2n+1)!!)^2 = Pi^2/8. - Daniel Suteu, Oct 31 2017

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

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

Original entry on oeis.org

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

A380570 Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2x - (2t - 1))Product_{t=1..n}(2x + (2*t - 1)).

Original entry on oeis.org

1, 4, -1, 16, -40, 9, 64, -560, 1036, -225, 256, -5376, 31584, -51664, 11025, 1024, -42240, 561792, -2764960, 4228884, -893025, 4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025, 16384, -1863680, 78926848, -1559683840, 14763100352, -61460460880, 87512357916

Offset: 0

Thomas Scheuerle, Jan 27 2025

Odd coefficients of x are excluded here because they are zero.

			Triangle begins:
 n \ k: 0        1        2          3          4           5          6
      x^0      x^2      x^4        x^6        x^8        x^10       x^12
[0]     1;
[1]     4,      -1;
[2]    16,     -40,       9;
[3]    64,    -560,    1036,      -225;
[4]   256,   -5376,   31584,    -51664,     11025;
[5]  1024,  -42240,  561792,  -2764960,   4228884,    -893025;
[6]  4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025;
     ...

Eric Weisstein's World of Mathematics, Hankel's Symbol.

Cf. A000302 (column 0).
Cf. A001818 (absolute values of main diagonal).
Cf. A001824 (1/4 of absolute values of second diagonal).
Cf. A001825 (1/16 of absolute values of second diagonal).
Cf. A380612 (row sums).
Cf. A008956.

T(n, k) = Vec(prod(k=1,n,2*x-(2*k-1))*prod(k=1,n,2*x+(2*k-1)))[1+2*k]

The Hankel symbol (x, n) is defined as (-1)^n*cos(Pi*x)*Gamma(1/2+n-x)*Gamma(1/2+n+x)/(Pi*n!) = (cos(Pi*x)/((-4)^n*n!))*Sum_{k=0..n} T(n, k)*x^(2*k)..
T(n, k) = A008956(n, k)*4^(n-k)*(-1)^k.
Sum_{k=0..n} T(n, k) = A380612(n) = -(-4)^n*Gamma(-1/2 + n)*Gamma(3/2 + n)/Pi.

cp's OEIS Frontend

A008956 Triangle of central factorial numbers |4^k t(2n+1,2n+1-2k)| read by rows (n>=0, k=0..n).

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A001824 Central factorial numbers: 1st subdiagonal of A008956.

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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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A380570 Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2*x - (2*t - 1))*Product_{t=1..n}(2*x + (2*t - 1)).

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A380570 Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2x - (2t - 1))Product_{t=1..n}(2x + (2*t - 1)).