A161736 -id:A161736 - 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

A133221 A001147 with each term repeated.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 15, 15, 105, 105, 945, 945, 10395, 10395, 135135, 135135, 2027025, 2027025, 34459425, 34459425, 654729075, 654729075, 13749310575, 13749310575, 316234143225, 316234143225, 7905853580625, 7905853580625, 213458046676875, 213458046676875

Offset: 0

N. J. A. Sloane, Oct 13 2007

nonn

Normally such sequences are excluded from the OEIS, but I have made an exception for this one because so many variants of it have occurred in recent submissions.
For n>=2, a(n) = product of odd positive integers <=(n-1). - Jaroslav Krizek, Mar 21 2009
a(n) is, for n>=3, the number of way to choose floor((n-1)/2) disjoint pairs of items from n-1 items. It is then a fortiori the size of the conjugacy class of the reversal permutation [n-1,n-2,n-3,...,3,2,1]=(1 n-1)(2 n-2)(3 n-3)... in the symmetric group on n-1 elements. - Karl-Dieter Crisman, Nov 03 2009

G. C. Greubel, Table of n, a(n) for n = 0..800

Cf. A006882, A055634.

Appears in A161736. - Johannes W. Meijer, Jun 18 2009

f[x_] := E^(x^2/2) + Sqrt[Pi/2]*Erfi[x/Sqrt[2]]; CoefficientList[ Series[f[x], {x, 0, 29}], x]*Range[0, 29]! (* Jean-François Alcover, Sep 25 2012, after Sergei N. Gladkovskii *)
Table[(n - 1 - Mod[n, 2])!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)
Table[((2 n + (-1)^n - 3)/2)!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)

a(n) = my(k = (2*n + (-1)^n - 3)/2); prod(i=0, (k-1)\2, k - 2*i) \\ Iain Fox, Dec 31 2017

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A133221(n): return Gauss_factorial(n-1, 2)
[A133221(n) for n in (0..29)]  # Peter Luschny, Oct 01 2012

E.g.f.: x*U(0)  where U(k)= 1 + (2*k+1)/(x - x^4/(x^3 + (2*k+2)*(2*k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 25 2012
G.f.: 1+x*G(0), where G(k)= 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
a(n) = (2*floor(n/2)-1)!! = (n-1-(n mod 2))!!. - Alois P. Heinz, Sep 24 2024

A161737 Numerators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 2, 16, 128, 2048, 32768, 262144, 2097152, 67108864, 2147483648, 17179869184, 137438953472, 2199023255552, 35184372088832, 281474976710656, 2251799813685248, 144115188075855872, 9223372036854775808, 73786976294838206464, 590295810358705651712, 9444732965739290427392

Offset: 1

Johannes W. Meijer, Jun 18 2009

For the definition of the BG2 matrix coefficients see A161736.

			sb(1) = 1; sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 1..830

Cf. A161736 and A161738, A050605, A007814 and A090739.

[Numerator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; x(1):=0: x(2):=1: for n from 2 to nmax-1 do x(n+1) := A050605(n-2) + x(n) + 3 od: for n from 1 to nmax do a(n) := 2^x(n) od: seq(a(n), n=1..nmax); # End program 1
nmax1 := 20; for n from 0 to nmax1 do y(2*n+1) := A090739(n); y(2*n) := A090739(n) od: z(1) := 0: z(2) := 1: for n from 3 to nmax1 do z(n) := z(n-1) + y(n-1) od: for n from 1 to nmax1 do a(n) := 2^z(n) od: seq(a(n), n=1..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012 and by Peter Luschny, Feb 13 2025
r := n -> Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> denom(simplify(r(n))):
seq(a(n), n = 1..20);  # Peter Luschny, Feb 12 2025

sb[1] = 1; sb[2] = 2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2;
Table[sb[n] // Numerator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

vector(20, n, n++; numerator((2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2))) \\ G. C. Greubel, Sep 26 2018

a(n) = numer(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161736(n) = denom(sb(n)).

a(n) = denominator(Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 12 2025

Offset set to 1 and a(1) = 1 prepended by Peter Luschny, Feb 13 2025

A161738 Sequence related to the column sums of the BG2 matrix.

Original entry on oeis.org

1, 1, 3, 15, 35, 315, 693, 9009, 19305, 328185, 692835, 14549535, 30421755, 760543875, 1579591125, 45808142625, 94670161425, 3124115327025, 6432002143875, 237984079323375, 488493636505875, 20028239096740875

Offset: 1

Johannes W. Meijer, Jun 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..667

Cf. A161736, A005408 and A093178.

[1] cat [(&*[2*n-2*k-3:k in [0..Floor(n/2 -1)]]): n in [2..50]]; // G. C. Greubel, Sep 26 2018

Table[Product[(2*n - 3 - 2*k), {k, 0, Floor[n/2 - 1]}], {n, 1, 50}] (* G. C. Greubel, Sep 26 2018 *)

for(n=1,50, print1(prod(k=0,floor(n/2 -1), 2*n-2*k-3), ", ")) \\ G. C. Greubel, Sep 26 2018

a(n) = product((2*n-3-2*k), k=0..floor(n/2-1)).
numer(a(n+2)/a(n+1)) = A005408(n) for n=>0.
denom(a(n+2)/a(n+1)) = A093178(n) for n=>0.

A278145 Denominator of partial sums of the m=1 member of an m-family of series considered by Hardy with value 4/Pi (see A088538).

Original entry on oeis.org

1, 8, 64, 1024, 16384, 131072, 1048576, 33554432, 1073741824, 8589934592, 68719476736, 1099511627776, 17592186044416, 140737488355328, 1125899906842624, 72057594037927936, 4611686018427387904, 36893488147419103232, 295147905179352825856, 4722366482869645213696

Offset: 0

Wolfdieter Lang, Nov 14 2016

The numerators seems to coincide with A161736(n+2).
Hardy considered the m-family of series H(m) = 1/m + (1/(m+1))*(1/2)^2 + (1/(m+2))*(1*3/(2*4))^2 + ... = Sum_{k>=0}(1/(m+k))*(risefac(1/2,k)/k!)^2, where risefac(x,m) = Product_{j=0..m-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 106, eq. (7.5.1) (with n=m).
The value of these series H(m) = (Gamma(m) / Gamma(m+1/2))^2 * Sum_{k = 0..m-1} (risefac(1/2,k)/k!)^2.
The present partial sums are for H(1) with value 1/Gamma(3/2)^2 = 4/Pi (A088538).

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.5.1), and references on p. 112 for Darling (1), p. 232, and Watson (5), p. 235.

Cf. A000165, A001147, A088538, A161736.

Table[Denominator@ Sum[(1/(k + 1)) (Pochhammer[1/2, k]/k!)^2, {k, 0, n}], {n, 0, 19}] (* or *)
Table[Denominator@ Sum[(1/(k + 1)) (Binomial[-1/2, k])^2, {k, 0, n}], {n, 0, 19}] (* or *)
Table[Denominator@ Sum[(1/(k + 1)) ((2 k - 1)!!/(2 k)!!)^2, {k, 0, n}], {n, 0, 19}] (* Michael De Vlieger, Nov 15 2016 *)

a(n)= denominator(r(n)) with the rationals r(n) = Sum_{k=0..n}(1/(k+1))*(risefac(1/2,k)/k!)^2 = Sum_{k=0..n} (1/(k+1))*(binomial(-1/2,k))^2 = Sum_{k=0..n}(1/(k+1))*((2*k-1)!!/(2*k)!!)^2 , with the rising factorial risefac(x,k) defined above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.

A280100 a(n) = 4^(2n) (n!)^3 * (n+1)!.

Original entry on oeis.org

1, 32, 12288, 21233664, 108716359680, 1304596316160000, 31560794080542720000, 1385645103312147578880000, 102160842176998016695664640000, 11916040631525048667382323609600000, 2097223151148408565459288955289600000000

Offset: 0

Daniel Suteu, Dec 25 2016

Daniel Suteu, Table of n, a(n) for n = 0..50

Cf. A134374, A161736, A278145.

PARI
```
a(n) = 4^(2*n) * (n!)^3 * (n+1)!;
```

a(n) ~ Pi/4 * A134374(n).
a(n) ~ Pi^2 * 2^(4*n+2) / exp(4*n+1) * n^(3*n+3/2) * (n+1)^(n+3/2).
Lim_{n->infinity} a(n) / ((2n+1)!)^2 = Pi/4.
a(n) / ((2n+1)!)^2 = A278145(n) / A161736(n+2).

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

Original entry on oeis.org

0, 1, 1, 4, 9, 64, 75, 256, 1225, 16384, 19845, 65536, 160083, 1048576, 1288287, 4194304, 41409225, 1073741824, 1329696225, 4294967296, 10667118605, 68719476736, 85530896451, 274877906944, 1371086188563, 17592186044416, 21972535073125, 70368744177664, 176021737014375

Offset: 0

Peter Luschny, Feb 11 2025

			r(n) = 0, 1, 1/2, 4/3, 9/16, 64/45, 75/128, 256/175, 1225/2048, ...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).

Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.

r := n -> (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2:
a := n -> numer(simplify(r(n))): seq(a(n), n = 0..28);
# Alternative:
r := n -> ifelse(n <= 1, n, (n - 1)/(n*r(n - 1))):

Join[{0}, Numerator[FoldList[(#2 - 1)/(#2*#) &, Range[30]]]] (* Paolo Xausa, Feb 14 2025 *)

Product_{k=1..n} a(k) = A380910(n) / A380909(n).
r(n) = (n - 1)/(n*r(n - 1)) for n > 1.
numerator(r(2*n)) = A161736(n).
numerator(r(2*n+1)) = A056982(n).
numerator(r(2*n+1))/4^n = A124399(n).
denominator(r(2*n-2)) = A161737(n).
denominator(r(2*n+1)) = A069955(n).
denominator(r(2*n+1))/(2*n+1) = A038534(n).
denominator(r(2*n+2))/2 = A278145(n).
denominator(r(2*n+2))/2^(2*n+1) = A001901(n).
r(n) ~ (2/Pi)^((-1)^n)*(1 - 1/(2*n) + 1/(8*n^2) + 1/(16*n^3) - 5/(128*n^4) - 23/(256*n^5) ...).

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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A133221 A001147 with each term repeated.

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A161737 Numerators of the column sums of the BG2 matrix.

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A161738 Sequence related to the column sums of the BG2 matrix.

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A278145 Denominator of partial sums of the m=1 member of an m-family of series considered by Hardy with value 4/Pi (see A088538).

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A280100 a(n) = 4^(2*n) * (n!)^3 * (n+1)!.

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A380949 a(n) = numerator(r(n)) where r(n) = (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2.

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A280100 a(n) = 4^(2n) (n!)^3 * (n+1)!.

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.