cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A046161 a(n) = denominator of binomial(2n,n)/4^n.

Original entry on oeis.org

1, 2, 8, 16, 128, 256, 1024, 2048, 32768, 65536, 262144, 524288, 4194304, 8388608, 33554432, 67108864, 2147483648, 4294967296, 17179869184, 34359738368, 274877906944, 549755813888, 2199023255552, 4398046511104, 70368744177664, 140737488355328, 562949953421312
Offset: 0

Views

Author

Eric W. Weisstein

Keywords

Comments

Also denominator of e(0,n) (see Maple line). - N. J. A. Sloane, Feb 16 2002
Denominator of coefficient of x^n in (1+x)^(k/2) or (1-x)^(k/2) for any odd integer k. - Michael Somos, Sep 15 2004
Numerator of binomial(2n,n)/4^n = A001790(n).
Denominators in expansion of sqrt(c(x)), c(x) the g.f. of A000108. - Paul Barry, Jul 12 2005
Denominator of 2^m*Gamma(m+3/4)/(Gamma(3/4)*Gamma(m+1)). - Stephen Crowley, Mar 19 2007
Denominator in expansion of Jacobi_P(n,1/2,1/2,x). - Paul Barry, Feb 13 2008
This sequence equals the denominators of the coefficients of the series expansions of (1-x)^((-1-2*n)/2) for all integer values of n; see A161198 for detailed information. - Johannes W. Meijer, Jun 08 2009
Numerators of binomial transform of 1, -1/3, 1/5, -1/7, 1/9, ... (Madhava-Gregory-Leibniz series for Pi/4): 1, 2/3, 8/15, 16/35, 128/315, 256/693, .... First differences are -1/3, -2/15, -8/105, -16/315, -128/3465, -256/9009, ... which contain the same numerators, negated. The second differences are 1/5, 2/35, 8/315, 16/1155, 128/15015, ... again with the same numerators. Second column: 2/3, -2/15, 2/35, -2/63, 2/99; see A000466(n+1) = A005563(2n+1). Third column: 8*(1/15, -1/105, 1/315, -1/693, ...), see A061550. See A173294 and A173296. - Paul Curtz, Feb 16 2010
0, 1, 5/3, 11/5, 93/35, 193/63, 793/231, ... = (0 followed by A120778(n))/A001790(n) is the binomial transform of 0, 1, -1/3, 1/5, -1/7, 1/9, ... . See A173755 and formula below. - Paul Curtz, Mar 13 2013
Numerator of power series of arcsin(x)/sqrt(1-x^2), centered at x=0. - John Molokach, Aug 02 2013
Denominators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n * Euler(2*n)*x^n/(2*n)), see A280442 for the numerators. - Johannes W. Meijer, Jan 05 2017
Denominators of Pochhammer(n+1, -1/2)/sqrt(Pi). - Adam Hugill, Sep 11 2022
a(n) is the denominator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Stefano Spezia, May 16 2023
4^n/binomial(2n,n) is the expected value of the number of socks that are randomly drawn out of a drawer of n different pairs of socks, when one sock is drawn out at a time until a matching pair is found (King and King, 2005). - Amiram Eldar, Jul 02 2023
a(n) is the denominator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding numerator is A001790(n). - Mohammed Yaseen, Jul 29 2023
a(n) is the numerator of Integral_{x=0..Pi/2} sin(x)^(2*n+1) dx. The corresponding denominator is A001803(n). - Mohammed Yaseen, Sep 22 2023

Examples

			sqrt(1+x) = 1 + (1/2)*x - (1/8)*x^2 + (1/16)*x^3 - (5/128)*x^4 + (7/256)*x^5 - (21/1024)*x^6 + (33/2048)*x^7 + ...
binomial(2n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...
The sequence e(0,n) begins 1, 3/2, 21/8, 77/16, 1155/128, 4389/256, 33649/1024, 129789/2048, 4023459/32768, ...

References

  • W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1.
  • B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.282).
  • Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 72.
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equations 6:14:5 - 6:14:9 at pages 50-51.

Links

Crossrefs

Cf. A001790, A001803, A002596, A005187, A067002, A072287, A142961.
Cf. A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n.
Cf. A000108, A000120, A000466, A001511, A005563, A048881, A060818, A061550, A120778, A173294, A173296, A173755, A280442.

Programs

  • Magma
    [Denominator(Binomial(2*n,n)/4^n): n in [0..30]]; // Vincenzo Librandi, Jul 18 2015
  • Maple
    e := proc(l,m) local k; add(2^(k-2*m)*binomial(2*m-2*k,m-k)* binomial(m+k, m) *binomial(k,l), k=l..m); end: seq(denom(e(0,n)), n = 0..24);
Z[0]:=0: for k to 30 do Z[k]:=simplify(1/(2-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(denom(coeff(gser, z, n)), n=-1..23); # Zerinvary Lajos, May 21 2008
A046161 := proc(n) option remember: if n = 0 then 1 else 2^A001511(n) * procname(n-1) fi: end: A001511 := proc(n): padic[ordp](2*n, 2) end: seq(A046161(n), n = 0..24); # Johannes W. Meijer, Nov 04 2012
A046161 := n -> 4^n/2^add(i,i=convert(n, base, 2)):
seq(A046161(n), n=0..24); # Peter Luschny, Apr 08 2014
  • Mathematica
    a[n_, m_] := Binomial[n - m/2 + 1, n - m + 1] - Binomial[n - m/2, n - m + 1]; s[n_] := Sum[ a[n, k], {k, 0, n}]; Table [Denominator[s[n]], {n, 0, 26}] (* Michele Dondi (bik.mido(AT)tiscalinet.it), Jul 11 2002 *)
Denominator[Table[Binomial[2n,n]/4^n,{n,0,30}]] (* Harvey P. Dale, Oct 29 2012 *)
Table[Denominator@LegendreP[2n,0],{n,0,24}] (* Andres Cicuttin, Jan 22 2018 *)
  • Maxima
    a(n) := denom(binomial(-1/2,n));
makelist(a(n),n,0,24); /* Peter Luschny, Nov 21 2012 */
  • PARI
    a(n)=if(n<0,0,denominator(binomial(2*n,n)/4^n)) /* Michael Somos, Sep 15 2004 */
  • PARI
    a(n)=my(s=n);while(n>>=1,s+=n);2^s \\ Charles R Greathouse IV, Apr 07 2012
  • PARI
    a(n)=denominator(I^-n*pollegendre(n,I/2)) \\ Charles R Greathouse IV, Mar 18 2017
  • Python
    def A046161(n): return 1<<(n<<1)-n.bit_count() # Chai Wah Wu, Nov 15 2022
  • Sage
    def A046161(n):
    A005187 = lambda n: A005187(n//2) + n if n > 0 else 0
    return 2^A005187(n)
[A046161(n) for n in (0..24)]  # Peter Luschny, Nov 16 2012

Formula

a(n) = 2^(2*n - 1 - A048881(n-1)), if n > 0.
a(n) = 2^A005187(n).
a(n) = 4^n/2^A000120(n). - Michael Somos, Sep 15 2004
a(n) = 2^A001511(n)*a(n-1) with a(0) = 1. - Johannes W. Meijer, Nov 04 2012
a(n) = denominator(binomial(-1/2,n)). - Peter Luschny, Nov 21 2012
a(n) = (0 followed by A120778(n)) + A001790(n). - Paul Curtz, Mar 13 2013
a(n) = 2^n*A060818(n). - Johannes W. Meijer, Jan 05 2017
a(n)/A001790(n) ~ sqrt(Pi*n) (King and King, 2005). - Amiram Eldar, Jul 02 2023

A142963 Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.

Original entry on oeis.org

1, 1, 2, 1, 10, 4, 1, 30, 72, 8, 1, 74, 516, 464, 16, 1, 166, 2584, 7016, 2864, 32, 1, 354, 10740, 64240, 84480, 17376, 64, 1, 734, 40008, 450280, 1321760, 949056, 104704, 128, 1, 1498, 139108, 2681296, 14713840, 24198976, 10223488, 629248, 256, 1, 3030, 462264, 14341992
Offset: 1

Views

Author

Wolfdieter Lang, Sep 15 2008

Keywords

Comments

Previous name: Table of coefficients of row polynomials of certain o.g.f.s.
The o.g.f.s G(k, x) for the k-family of sequences S(k, n):= Sum_{p=0..n} p^k*binomial(2*p, p)*binomial(2*(n-p), n-p), k=0,1,... (convolution of two sequences involving the central binomial coefficients) are 1/(1-4*x) for k=0 and 2*x*P(k, x)/(1-4*x)^(k+1) for k=1,2,..., with the row polynomials P(k, x) = Sum_{m=0..k-1} a(n,m)*x^m).
The author was led to compute the sums S(k, n) by a question asked by M. Greiter, Jun 27 2008.
In order to keep the index k>=1 of Sigma(k, n) also for the polynomials P(k, x), their degree is then k-1.

Examples

			Triangle starts:
[1]
[1,   2]
[1,  10,     4]
[1,  30,    72,      8]
[1,  74,   516,    464,      16]
[1, 166,  2584,   7016,    2864,     32]
[1, 354, 10740,  64240,   84480,  17376,     64]
[1, 734, 40008, 450280, 1321760, 949056, 104704, 128]
...
P(3,x) = 1+10*x+4*x^2.
G(3,x) = 2*x*(1+10*x+4*x^2)/(1-4*x)^4.

Links

Crossrefs

Left hand column sequences 2*A142964, 4*A142965, 8*A142966, 16*A142968.
Row sums A142967.
From Johannes W. Meijer, Feb 20 2009: (Start)
A156919 and this sequence can be mapped onto A156920.
Cf. A156921, A156925, A156927, A156933.
Right hand column sequences 2^n*A000340, 2^n*A156922, 2^n*A156923, 2^n*A156924. (End)
Cf. A142961, A142962.

Programs

  • Maple
    A142963 := proc(n,m): if n=m+1 then 2^(n-1); elif m=0 then 1 ; elif m<0 or m>n-1 then 0; else (m+1)*procname(n-1, m)+(4*n-4*m-2)*procname(n-1, m-1); end if; end proc: seq(seq(A142963(n,m), m=0..n-1), n=1..9); # Johannes W. Meijer, Sep 28 2011
# Alternatively (assumes offset 0):
p := (n,x) -> (1/2)*add(Stirling2(n+1,k+1)*x^k*(1-4*x)^(n-k)*(2*k+2)!/(k+1)!, k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n,x), x) od;
# Peter Luschny, Jun 18 2017
  • Mathematica
    t[, 0] = 1; t[n, m_] /; m == n-1 := 2^m; t[n_, m_] := (m+1)*t[n-1, m] + (4*n-4*m-2)*t[n-1, m-1]; Table[t[n, m], {n, 1, 10}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jun 21 2013, after Johannes W. Meijer *)

Formula

G(k, x) = Sum_{p=0..k} S2(k, p)*((2*p)!/p!)*x^p/(1-4*x)^(p+1), k >= 0 (here k >= 1), with the Stirling2 triangle S2(k, p):=A048993(k, p). (Proof from the product of the o.g.f.s of the two convoluted sequences and the normal ordering (x^d_x)^k = Sum_{p=0..k} S2(k, p)*x^p*d_x^p, with the derivative operator d_x.)
a(k,m) = [x^m]P(k, x) = [x^m] ((1-4*x)^(k+1))*G(k,x)/(2*x), k>=1, m=0,1,...,k-1.
For the triangle coefficients the following relation holds: T(n,m) = (m+1)*T(n-1,m) + (4*n-4*m-2)*T(n-1,m-1) with T(n,m=0) = 1 and T(n,m=n-1) = 2^(n-1), n >= 1 and 0 <= m <= n-1. - Johannes W. Meijer, Feb 20 2009
From Peter Bala, Jan 18 2018: (Start)
(x*d/dx)^n (1/(sqrt(1 - 4*x)) = 2*x*P(n,x)/sqrt(1 - 4*x)^(n+1/2) for n >= 1.
x*P(n,x)/(1 - 4*x)^(n+1/2) = (1/2)*Sum_{k >= 1} binomial(2*k,k)* k^n*x^k for n >= 1.
P(n+1,x) = ((4*n - 2)*x + 1)*P(n,x) - x*(4*x - 1)*d/dx(P(n,x)).
Hence the polynomial P(n,x) has all real zeros by Liu et al., Theorem 1.1, Corollary 1.2. (End)

Extensions

Minor edits by Johannes W. Meijer, Sep 28 2011
A more precise name by Peter Luschny, Jun 18 2017
Name reformulated with offset corrected, edited by Wolfdieter Lang, Aug 23 2019

A142962 Scaled convolution of (n^3)*A000984(n) with A000984(n).

Original entry on oeis.org

4, 26, 81, 184, 350, 594, 931, 1376, 1944, 2650, 3509, 4536, 5746, 7154, 8775, 10624, 12716, 15066, 17689, 20600, 23814, 27346, 31211, 35424, 40000, 44954, 50301, 56056, 62234, 68850, 75919, 83456, 91476, 99994, 109025, 118584, 128686, 139346, 150579
Offset: 1

Views

Author

Wolfdieter Lang, Sep 15 2008

Keywords

Comments

S(3,n) := Sum_{j=0..n} j^3*binomial(2*j,j)*binomial(2*(n-j),n-j).
a(n) = 2^3*S(3,n)/4^n, n >= 1.
O.g.f. for S(3,n) is G(k=3,x). See triangle A142963 for the general G(k,x) formula.
The author was led to compute such sums by a question asked by M. Greiter, Jun 27 2008.

Links

Crossrefs

Cf. A142961 triangle: row k=3: [3, 5], with the row polynomial 3+5*n.
Cf. A049451 (scaled k=2 case).

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x (4 + 10 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Jul 02 2023 *)

Formula

a(n) = n^2*(3+5*n)/2.
a(n) = (2^3)*S(3,n)/4^n with the convolution S(3,n) defined above.
G.f.: x*(4+10*x+x^2)/(1-x)^4. - Joerg Arndt, Jul 02 2023
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