A002455 -id:A002455 - cp's OEIS frontend

A001819 Central factorial numbers: second right-hand column of triangle A008955.

0, 1, 5, 49, 820, 21076, 773136, 38402064, 2483133696, 202759531776, 20407635072000, 2482492033152000, 359072203696128000, 60912644957448192000, 11977654199703478272000, 2702572249389834608640000

Offset: 0

N. J. A. Sloane

Coefficient of x^2 in Product_{k=0..n}(x + k^2). - Ralf Stephan, Aug 22 2004
p divides a(p-1) for prime p > 3. p divides a((p-1)/2) for prime p > 3. For prime p, p^2 divides a(n) for n > 2*p+1. - Alexander Adamchuk, Jul 11 2006; last comment corrected by Michel Marcus, May 20 2020
The ratio a(n)/A001044(n) is the partial sum of the reciprocals of squares. E.g., a(4)/A001044(4) = 820/576 = 1/1 + 1/4 + 1/9 + 1/16. - Pierre CAMI, Oct 30 2006
a(n) is the (n-1)-st elementary symmetric function of the squares of the first n numbers. - Anton Zakharov, Nov 06 2016
Primes p such that p^2 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019

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..253 (terms 0..50 from T. D. Noe)
Takao Komatsu, Convolution identities of poly-Cauchy numbers with level 2, arXiv:2003.12926 [math.NT], 2020.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
Index entries for sequences related to factorial numbers

Cf. A000254, A001044, A002455, A007406, A007407, A066989, A024167, A142995.
Second right-hand column of triangle A008955.
Equals row sums of A162990(n)/(n+1)^2 for n >= 1.

Table[Sum[1/i^2,{i,1,n}]/Product[1/i^2,{i,1,n}],{n,1,40}] (* Alexander Adamchuk, Jul 11 2006 *)
Table[n!^2*HarmonicNumber[n, 2], {n, 0, 15}] (* Jean-François Alcover, May 09 2012, after Joe Keane *)

a(n)=n!^2*sum(k=1,n,1/k^2) \\ Charles R Greathouse IV, Nov 06 2016

a_n = (n!)^2 * Sum_{k=1..n} 1/k^2. - Joe Keane (jgk(AT)jgk.org)
a(n) ~ (1/3)*Pi^3*n*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
Sum_{n>=0} a(n)*x^n/n!^2 = polylog(2, x)/(1-x). - Vladeta Jovovic, Jan 23 2003
a(n) = Sum_{i=1..n} 1/i^2 / Product_{i=1..n} 1/i^2. - Alexander Adamchuk, Jul 11 2006
a(0) = 0, a(n) = a(n-1)*n^2 + A001044(n-1). E.g., a(1) = 0*1 + 1 = 1 since A001044(0) = 1; a(2) = 1*2^2 + 1 = 5 since A001044(1) = 1; a(3) = 5*3^2 + 4 = 49 since A001044(2) = 4; and so on. - Pierre CAMI, Oct 30 2006
Recurrence: a(0) = 0, a(1) = 1, a(n+1) = (2*n^2 + 2*n + 1)*a(n) - n^4*a(n-1). The sequence b(n) = n!^2 satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(1 - 1^4/(5 - 2^4/(13 - 3^4/(25 - ... -(n-1)^4/((2*n^2 - 2*n + 1)))))), leading to the infinite continued fraction expansion zeta(2) = 1/(1-1^4/(5 - 2^4/(13 - 3^4/(25 - ... - n^4/((2*n^2 + 2*n + 1) - ...))))). Compare with A142995. Compare also with A024167 and A066989. - Peter Bala, Jul 18 2008
a(n)/(n!)^2 -> zeta(2) = A013661 as n -> infinity, rewriting the Keane formula. - Najam Haq (njmalhq(AT)yahoo.com), Jan 13 2010
a(n) = s(n+1,2)^2 - 2*s(n+1,1)*s(n+1,3), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 03 2012

Minor edits by Vaclav Kotesovec, Jan 28 2015

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)

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

A002701 Coefficients for numerical differentiation.

Original entry on oeis.org

1, 1, 7, 41, 479, 59, 266681, 63397, 514639, 178939, 10410343, 18500393, 40799043101, 1411432849, 6620481151, 48409924397, 238357395880861, 339716530787, 86364397717734821, 421950627598601, 222226462279, 15392144025383

Offset: 2

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).

H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135.
H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135. [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21 (see numerators of numbers named N(3,2k+1) and N(4,2k)).

Cf. A002455 (central factorial numbers), A002702 (denominators).

a(n) = numer(6 * A002455(n) / 2^(2*n) * (2*n + 3)!). - Sean A. Irvine, Jun 10 2014

More terms from Sean A. Irvine, Jun 10 2014

A002702 Coefficients for numerical differentiation.

Original entry on oeis.org

1, 4, 120, 3024, 151200, 79200, 1513512000, 1513512000, 51459408000, 74662922880, 18068427336960, 133196739984000, 1215553449093984000, 173650492727712000, 3357242859402432000, 101013513093196704000, 2043503369875369321920000

Offset: 2

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).

H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135.
H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135. [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21 (see denominators of numbers named N(3,2k+1)).

Cf. A002455 (central factorial numbers), A002701 (numerators).

a(n) = den(6 * A002455(n) / 2^(2*n) * (2*n + 3)!). - Sean A. Irvine, Jun 10 2014

More terms from Sean A. Irvine, Jun 10 2014

cp's OEIS Frontend

A001819 Central factorial numbers: second right-hand column of triangle A008955.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A001824 Central factorial numbers: 1st subdiagonal of A008956.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A001825 Central factorial numbers: 2nd subdiagonal of A008956.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple

Formula

A002701 Coefficients for numerical differentiation.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A002702 Coefficients for numerical differentiation.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions