A001824 -id:A001824 - 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

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)

A001825 Central factorial numbers: 2nd subdiagonal of A008956.

Original entry on oeis.org

1, 35, 1974, 172810, 21967231, 3841278805, 886165820604, 261042753755556, 95668443268795341, 42707926241367380631, 22821422608929422854674, 14384681946935352617964750, 10562341153570752891930640875

Offset: 0

N. J. A. Sloane

nonn

			(arcsin x)^5 = x^5 + 5/6*x^7 + 47/72*x^9 + 1571/3024*x^11 + ...

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

Cf. A001824, A002455, A049033.

Right-hand column 3 in triangle A008956.

Table[(2*n+5)!/5! * SeriesCoefficient[ArcSin[x]^5,{x,0,2*n+5}], {n,0,20}] (* Vaclav Kotesovec, Feb 23 2015 *)

E.g.f.: (arcsin x)^5; that is, a_k is the coefficient of x^(2*k+5) in (arcsin x)^5 multiplied by (2*k+5)! and divided by 5!. - Joe Keane (jgk(AT)jgk.org)
(-1)^(n-2)*a(n-2) is the coefficient of x^4 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22 2002
a(n) = det(V(i+3,j+2), 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
a(n) = (12*n^2 + 12*n + 11)*a(n-1) - (4*n^2 + 3)*(12*n^2 + 1)*a(n-2) + (2*n - 1)^6*a(n-3). - Vaclav Kotesovec, Feb 23 2015
a(n) ~ Pi^4 * n^(2*n+4) * 2^(2*n-2) / (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

A002555 Denominators of coefficients for numerical differentiation.

Original entry on oeis.org

1, 24, 5760, 322560, 51609600, 13624934400, 19837904486400, 2116043145216, 20720294477955072, 15747423803245854720, 131978409017679544320, 72852081777759108464640, 151532330097738945606451200, 2828603495157793651320422400, 19687080326298243813190139904000

Offset: 1

N. J. A. Sloane

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).

Ruperto Corso, Table of n, a(n) for n = 1..387
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 23.

Cf. A001824, A002554.

with(combinat): a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n-1))*(-1)^(n-1)/(2^(2*n-3)*(2*n)!): seq(denom(a(n)), n=1..20); # Ruperto Corso, Dec 15 2011

a(n) is the denominator of (-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n)!*2^(2n-3)), where Cn{1^2..(2n+1)^2} is equal to 1 when n=0, otherwise, it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,..,n. - Ruperto Corso, Dec 15 2011

a(n) = denominator(A001824(n-1)*(-1)^(n-1)/(2^(2*n-3)*(2*n)!)). - Sean A. Irvine, Mar 29 2014

More terms from Ruperto Corso, Dec 15 2011

A291656 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n-1)!!)^k * Sum_{i=1..n} 1/(2*i-1)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 10, 23, 4, 0, 1, 28, 259, 176, 5, 0, 1, 82, 3527, 12916, 1689, 6, 0, 1, 244, 51331, 1213136, 1057221, 19524, 7, 0, 1, 730, 762743, 123296356, 885533769, 128816766, 264207, 8, 0, 1, 2188, 11406979, 12820180976, 809068942341, 1179489355164, 21878089479, 4098240, 9

Offset: 0

Seiichi Manyama, Aug 28 2017

			Square array begins:
  0,   0,     0,       0,         0, ...
  1,   1,     1,       1,         1, ...
  2,   4,    10,      28,        82, ...
  3,  23,   259,    3527,     51331, ...
  4, 176, 12916, 1213136, 123296356, ...

Seiichi Manyama, Antidiagonals n = 0..54, flattened

Columns k=0-5 give: A001477, A004041(n+1), A001824(n+1), A291585, A291586, A291587.
Rows n=0-2 give: A000004, A000012, A034472.
Main diagonal gives A291676.
Cf. A291556.

T(0,k) = 0, T(1,k) = 1 and T(n+1, k) = ((2*n-1)^k+(2*n+1)^k) * T(n, k) - (2*n-1)^(2*k) * T(n-1, k).

A335090 a(n) = ((2n+1)!!)^2 (Sum_{k=1..n} 1/(2*k+1)^2).

Original entry on oeis.org

0, 1, 34, 1891, 164196, 20760741, 3616621254, 832001250375, 244557191709000, 89472598178279625, 39886085958271670250, 21288783013213520392875, 13405493416599700058947500, 9835107221539462476348118125, 8316889511005794888839427108750, 8030850428074789829954674314399375

Offset: 0

Seiichi Manyama, Sep 11 2020

nonn

Column k=2 of A335095.

Cf. A001824, A334670, A335091, A335092.

a[n_] := ((2*n + 1)!!)^2 * Sum[1/(2*k + 1)^2, {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Apr 29 2021 *)

{a(n) = prod(k=1, n, 2*k+1)^2*sum(k=1, n, 1/(2*k+1)^2)}

{a(n) = if(n<2, n, (8*n^2+2)*a(n-1)-(2*n-1)^4*a(n-2))}

a(n) = (8*n^2+2) * a(n-1) - (2*n-1)^4 * a(n-2) for n>1.

a(n) ~ (Pi^2/8 - 1) * 2^(2*n + 3) * n^(2*n + 2) / exp(2*n). - Vaclav Kotesovec, Sep 25 2020

A002554 Numerators of coefficients for numerical differentiation.

Original entry on oeis.org

1, -5, 259, -3229, 117469, -7156487, 2430898831, -60997921, 141433003757, -25587296781661, 51270597630767, -6791120985104747, 3400039831130408821, -15317460638921852507, 25789165074168004597399, -1550286106708510672406629, 24823277118070193095631689

Offset: 1

N. J. A. Sloane

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).

Ruperto Corso, Table of n, a(n) for n = 1..387
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 23.

Cf. A001824, A002555.

with(combinat):
a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n-1))*(-1)^(n-1)/(2^(2*n-3)*(2*n)!):
seq(numer(a(n)), n=1..20); # Ruperto Corso, Dec 15 2011

a(n) is the numerator of (-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n)!*2^(2n-3)), where Cn{1^2..(2n+1)^2} equals 1 when n=0, otherwise it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,...,n. - Ruperto Corso, Dec 15 2011

a(n) = numerator(A001824(n-1)*(-1)^(n-1)/(2^(2*n-3)*(2*n)!)). - Sean A. Irvine, Mar 29 2014

Corrected and extended by Ruperto Corso, Dec 15 2011

A291587 a(n) = ((2n-1)!!)^5 * Sum_{i=1..n} 1/(2*i-1)^5.

Original entry on oeis.org

0, 1, 244, 762743, 12820180976, 757031629267449, 121921454556651769524, 45268703999809586294371407, 34375967164840303438628549400000, 48808991831991566280900452880679940625, 120855944455445379138034328603009420077012500

Offset: 0

Seiichi Manyama, Aug 27 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..105

Cf. A001147, A099827.

Cf. A004041, A001824, A291585, A291586.

Table[(2*n-1)!!^5 * Sum[1/(2*i-1)^5, {i, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(0) = 0, a(1) = 1, a(n+1) = ((2*n-1)^5+(2*n+1)^5)*a(n) - (2*n-1)^10*a(n-1) for n > 0.

a(n) ~ 31*Zeta(5) * 2^(5*n-5/2) * n^(5*n) / exp(5*n). - Vaclav Kotesovec, Aug 27 2017

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

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A001825 Central factorial numbers: 2nd subdiagonal of A008956.

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

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A002555 Denominators of coefficients for numerical differentiation.

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A291656 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n-1)!!)^k * Sum_{i=1..n} 1/(2*i-1)^k.

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A335090 a(n) = ((2*n+1)!!)^2 * (Sum_{k=1..n} 1/(2*k+1)^2).

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A335090 a(n) = ((2n+1)!!)^2 (Sum_{k=1..n} 1/(2*k+1)^2).

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)).