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 1-10 of 12 results. Next

A084543 a(2,n) as defined in A003148.

Original entry on oeis.org

3, -9, 69, -531, 6147, -73665, 1143045, -18456795, 359420355, -7279744185, 170639259525, -4151789871075, 113815089771075, -3231091019581425, 101428235414230725, -3289812808335928875, 116369085609950047875, -4244245621899589931625, 167038144384492533277125
Offset: 0

Views

Author

R. J. Mathar, Jul 01 2003

Keywords

Programs

  • Maple
    A003148 := proc(m::integer,n::integer) doublefactorial(2*n+2*m+1)/(2*m+1)*simplify(hypergeom([ -n,m+1/2],[m+3/2],2)) ; end proc:
A084543 := proc(n::integer) A003148(2,n) ; end proc:
seq(A084543(n),n=0..40) : # R. J. Mathar, Apr 25 2006
  • Mathematica
    a[m_, n_] := (2n + 2m + 1)!!/(2m + 1)*Hypergeometric2F1[-n, m + 1/2, m + 3/2, 2];
Table[a[2, n], {n, 0, 20}] (* Jean-François Alcover, Jul 31 2023, after R. J. Mathar *)

Formula

D-finite with recurrence a(n) + 3*a(n-1) - 2*(2*n+3)*(n-1)*a(n-2) = 0. - R. J. Mathar, Mar 12 2013

Extensions

More terms from R. J. Mathar, Apr 25 2006

A077568 a(1,n) as defined in A003148.

Original entry on oeis.org

1, -1, 11, -39, 633, -4065, 86355, -818055, 21370545, -270059265, 8348125275, -132575387175, 4724044288425, -90632895177825, 3652562288650275, -82321915303002375, 3698358581066774625, -95898903720429434625, 4748433998702431912875, -139390494822185358126375
Offset: 0

Views

Author

R. J. Mathar, Jul 01 2003

Keywords

Crossrefs

Cf. A003148, A084543.

Programs

  • Maple
    A003148 := proc(m::integer,n::integer) RETURN( doublefactorial(2*n+2*m+1)/(2*m+1)*simplify(hypergeom([ -n,m+1/2],[m+3/2],2))) ; end proc:
A077568 := proc(n::integer) A003148(1,n) ; end proc:
for n from 0 to 20 do print(A077568(n)); od: # R. J. Mathar, Apr 25 2006
  • Mathematica
    a[m_, n_] := (2n + 2m + 1)!!/(2m + 1)*Hypergeometric2F1[-n, m + 1/2, m + 3/2, 2];
Table[a[1, n], {n, 0, 15}] (* Jean-François Alcover, Jul 31 2023, after R. J. Mathar *)

Formula

Conjecture: a(n) +a(n-1) -2*(2*n+1)*(n-1)*a(n-2)=0. - R. J. Mathar, May 26 2016

Extensions

More terms from R. J. Mathar, Apr 25 2006

A049606 Largest odd divisor of n!.

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 42567525, 638512875, 638512875, 10854718875, 97692469875, 1856156927625, 9280784638125, 194896477400625, 2143861251406875, 49308808782358125, 147926426347074375, 3698160658676859375
Offset: 0

Views

Author

N. J. A. Sloane, Feb 05 2000

Keywords

Comments

Original name: Denominator of 2^n/n!.
For positive n, a(n) equals the numerator of the permanent of the n X n matrix whose (i,j)-entry is cos(i*Pi/3)*cos(j*Pi/3) (see example below). - John M. Campbell, May 28 2011
a(n) is also the number of binomial heaps with n nodes. - Zhujun Zhang, Jun 16 2019
a(n) is the number of 2-Sylow subgroups of the symmetric group S_n (see the Mathematics Stack Exchange link below). - Jianing Song, Nov 11 2022

Examples

			From _John M. Campbell_, May 28 2011: (Start)
The numerator of the permanent of the following 5 X 5 matrix is equal to a(5):
|  1/4  -1/4  -1/2  -1/4   1/4 |
| -1/4   1/4   1/2   1/4  -1/4 |
| -1/2   1/2    1    1/2  -1/2 |
| -1/4   1/4   1/2   1/4  -1/4 |
|  1/4  -1/4  -1/2  -1/4   1/4 | (End)

Links

Crossrefs

Numerators of 2^n/n! give A001316. Cf. A000680, A008977, A139541.
Factor of A160481. - Johannes W. Meijer, May 24 2009
Equals A003148 divided by A123746. - Johannes W. Meijer, Nov 23 2009
Different from A160624.
Cf. A011371.

Programs

  • Magma
    [ Denominator(2^n/Factorial(n)): n in [0..25] ]; // Klaus Brockhaus, Mar 10 2011
  • Maple
    f:= n-> n! * 2^(add(i,i=convert(n,base,2))-n); # Peter Luschny, May 02 2009
seq (denom (coeff (series(1/(tanh(t)-1), t, 30), t, n)), n=0..25); # Peter Luschny, Aug 04 2011
seq(numer(n!/2^n), n=0..100); # Robert Israel, Jul 23 2015
  • Mathematica
    Denominator[Table[(2^n)/n!,{n,0,40}]] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011*)
Table[Last[Select[Divisors[n!],OddQ]],{n,0,30}] (* Harvey P. Dale, Jul 24 2016 *)
Table[n!/2^IntegerExponent[n!,2], {n,1,30}] (* Clark Kimberling, Oct 22 2016 *)
  • PARI
    A049606(n)=local(f=n!);f/2^valuation(f,2); \\ Joerg Arndt, Apr 22 2011
(Python 3.10+)
from math import factorial
def A049606(n): return factorial(n)>>n-n.bit_count() # Chai Wah Wu, Jul 11 2022

Formula

a(n) = Product_{k=1..n} A000265(k).
a(n) = A000265(A000142(n)). - Reinhard Zumkeller, Apr 09 2004
a(n) = numerator(2*Sum_{i>=1} (-1)^i*(1-zeta(n+i+1)) * (Product_{j=1..n} i+j)). - Gerry Martens, Mar 10 2011
a(n) = denominator([t^n] 1/(tanh(t)-1)). - Peter Luschny, Aug 04 2011
a(n) = n!/2^A011371(n). - Robert Israel, Jul 23 2015
From Zhujun Zhang, Jun 16 2019: (Start)
a(n) = n!/A060818(n).
E.g.f.: Product_{k>=0} (1 + x^(2^k) / 2^(2^k - 1)).
(End)
log a(n) = n log n - (1 + log 2)n + Θ(log n). - Charles R Greathouse IV, Feb 12 2022

Extensions

New name (from Amarnath Murthy) by Charles R Greathouse IV, Jul 23 2015

A024199 a(n) = (2n-1)!! * Sum_{k=0..n-1}(-1)^k/(2k+1).

Original entry on oeis.org

0, 1, 2, 13, 76, 789, 7734, 110937, 1528920, 28018665, 497895210, 11110528485, 241792844580, 6361055257725, 163842638377950, 4964894559637425, 147721447995130800, 5066706567801827025, 171002070002301095250, 6548719685561840296125, 247199273204273879989500
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

(2*n + 1)!!/a(n+1), n>=0, is the n-th approximant for William Brouncker's continued fraction for 4/Pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... ))) See the C. Brezinski and J.-P. Delahaye references given under A142969 and A142970, respectively. The double factorials (2*n + 1)!! = A001147(n+1) enter. - Wolfdieter Lang, Oct 06 2008

Examples

			a(3) = (2*3 - 1)!! * Sum_{k=0..2} (-1)^k/(2k + 1) = 5!! * (1/(2*0 + 1) - 1/(2*1 + 1) + 1/(2*2 + 1)) = 5*3*1*(1/1 - 1/3 + 1/5) = 15 - 5 + 3 = 13. Notice that the first factor always cancels the common denominator of the sum. - _Michael B. Porter_, Jul 22 2016

References

  • A. E. Jolliffe, Continued Fractions, in Encyclopaedia Britannica, 11th ed., pp. 30-33; see p. 31.

Links

Crossrefs

Cf. A004041, A000407.
From Johannes W. Meijer, Nov 12 2009: (Start)
Cf. A007509 and A025547.
Equals first column of A167584.
Equals row sums of A167591.
Equals first right hand column of A167594.
(End)
Cf. A167576 and A135457.

Programs

  • Magma
    [0] cat [ n le 2 select (n) else 2*Self(n-1)+(2*n-3)^2*Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Feb 17 2015
  • Maple
    a := proc(n) option remember; if n=0 then 0 elif n=1 then 1 else 2*a(n-1)+(2*n-3)^2* a(n-2) fi end: seq(a(n), n=0..20); # Peter Luschny, Nov 16 2016 after N. J. A. Sloane
  • Mathematica
    f[k_] := (2 k - 1) (-1)^(k + 1)
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}]    (* A024199 signed *)
(* Clark Kimberling, Dec 30 2011 *)
RecurrenceTable[{a[n+1] == 2*a[n] + (2*n-1)^2*a[n-1],a[0] == 0, a[1] == 1},a,{n,0,20}] (* Vaclav Kotesovec, Mar 18 2014 *)
CoefficientList[Series[Pi/4/Sqrt[1-2*x] - 1/2*Log[2*x+Sqrt[4*x^2-1]]/Sqrt[2*x-1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 18 2014 *)

Formula

a(n) = s(1)s(2)...s(n)(1/s(1) - 1/s(2) + ... + c/s(n)) where c=(-1)^(n+1) and s(k) = 2k-1 for k = 1, 2, 3, ... (was previous definition). - Clark Kimberling
D-finite with recurrence a(0) = 0, a(1) = 1, a(n+1) = 2*a(n) + (2*n-1)^2*a(n-1). - N. J. A. Sloane, Jul 19 2002
a(n) + A024200(n) = A001147(n) = (2n-1)!!. - Max Alekseyev, Sep 23 2007
a(n)/A024200(n) -> Pi/(4-Pi) as n -> oo. - Max Alekseyev, Sep 23 2007
From Wolfdieter Lang, Oct 06 2008: (Start)
E.g.f. for a(n+1), n>=0: (sqrt(1-2*x)+arcsin(2*x)*sqrt(1+2*x)/2)/((1-4*x^2)^(1/2)*(1-2*x)). From the recurrence, solving (1-4*x^2)y''(x)-2*(8*x+1)*y'(x)-9*y=0 with inputs y(0)=1, y'(0)=2.
a(n+1) = A003148(n) + A143165(n), n>=0 (from the two terms of the e.g.f.). (End)
From Johannes W. Meijer, Nov 12 2009: (Start)
a(n) = (-1)^(n-1)*(2*n-3)!! + (2*n-1)*a(n-1) with a(0) = 0.
a(n) = (2*n-1)!!*sum((-1)^(k)/(2*k+1), k=0..n-1)
(End)
E.g.f.: Pi/4/sqrt(1-2*x) - 1/2*log(2*x+sqrt(4*x^2-1))/sqrt(2*x-1). - Vaclav Kotesovec, Mar 18 2014
a(n) ~ Pi * 2^(n-3/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 18 2014
a(n) = (2*H(n+1/2)-Gamma(n+1/2))*2^(n-2)*sqrt(Pi) with H(x) the Hadamard factorial (see the link section). - Cyril Damamme, Jul 19 2015
a(n) = A135457(n) - (-1)^n A001147(n-1). - Cyril Damamme, Jul 19 2015
a(n) = (Pi + (-1)^n*(Psi(n/2 + 1/4) - Psi(n/2 + 3/4)))*Gamma(n+1/2)*2^(n-2)/sqrt(Pi). - Robert Israel, Jul 20 2015
a(n) = A167576(n) - A135457(n). - Cyril Damamme, Jul 22 2015
a(n)/A001147(n) -> Pi/4 as n -> oo. - Daniel Suteu, Jul 21 2016
From Peter Bala, Nov 15 2016: (Start)
Conjecture: a(n) = 1/2*Sum_{k = 0..2*n-1} (-1)^(n-k+1)*k!*(2*n - 2*k - 3)!!, where the double factorial of an odd integer (positive or negative) may be defined in terms of the gamma function as (2*N - 1)!! = 2^((N+1)/2)*Gamma(N/2 + 1)/sqrt(Pi).
E.g.f. 1/2*arcsin(2*x)/sqrt(1 - 2*x) = x + 2*x^2/2! + 13*x^3/3! + 76*x^4/4! + .... (End)

Extensions

Edited by N. J. A. Sloane, Jul 19 2002
New name from Cyril Damamme, Jul 19 2015

A167591 A triangle related to the a(n) formulas of the rows of the ED4 array A167584.

Original entry on oeis.org

1, 4, -2, 12, -8, 9, 32, -16, 120, -60, 80, 0, 952, -768, 525, 192, 160, 5664, -5008, 12396, -5670, 448, 896, 27888, -20672, 162740, -133128, 72765, 1024, 3584, 120064, -46720, 1537216, -1562464, 2557296, -1081080, 2304, 12288, 467712, 76800
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED4 array A167584.
The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 4*n - 2.
Row 3: a(n) = 12*n^2 - 8*n + 9.
Row 4: a(n) = 32*n^3 - 16*n^2 + 120*n - 60.
Row 5: a(n) = 80*n^4 + 0*n^3 + 952*n^2 - 768*n + 525.
Row 6: a(n) = 192*n^5 + 160*n^4 + 5664*n^3 - 5008*n^2 + 12396*n - 5670.
Row 7: a(n) = 448*n^6 + 896*n^5 + 27888*n^4 - 20672*n^3 + 162740*n^2 - 133128*n + 72765.
Row 8: a(n) = 1024*n^7 + 3584*n^6 + 120064*n^5 - 46720*n^4 + 1537216*n^3 - 1562464*n^2 + 2557296*n - 1081080.
Row 9: a(n) = 2304*n^8 + 12288*n^7 + 467712*n^6 + 76800*n^5 + 11589216*n^4 - 12058368*n^3 + 47963568*n^2 - 38278080*n + 18243225.
Row 10: a(n) = 5120*n^9 + 38400*n^8 + 1686528*n^7 + 1540608*n^6 + 73898880*n^5 - 66179520*n^4 + 631348672*n^3 - 669559008*n^2 + 869709780*n - 344594250.

Crossrefs

A167584 is the ED4 array.
A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the ED4 array.
A001787, A167592, A167593, A168307 and A168308 equal the first five left hand triangle columns.
A001193 equals the first right hand triangle column.
A024199 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Extensions

Comment and formulas added by Johannes W. Meijer, Nov 23 2009

A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

Original entry on oeis.org

1, 6, -1, 20, 0, 3, 56, 28, 98, -15, 144, 192, 1080, -48, 105, 352, 880, 7568, 2024, 6534, -945, 832, 3328, 40976, 31616, 132444, -8112, 10395, 1920, 11200, 187488, 274480, 1593960, 286900, 972162, -135135, 4352, 34816, 761600, 1784320, 13962848
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED3 array A167572.
The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 6*n - 1.
Row 3: a(n) = 20*n^2 + 0*n + 3.
Row 4: a(n) = 56*n^3 + 28*n^2 + 98*n - 15.
Row 5: a(n) = 144*n^4 + 192*n^3 + 1080*n^2 - 48*n + 105.
Row 6: a(n) = 352*n^5 + 880*n^4 + 7568*n^3 + 2024*n^2 + 6534*n - 945.
Row 7: a(n) = 832*n^6 + 3328*n^5 + 40976*n^4 + 31616*n^3 + 132444*n^2 - 8112*n + 10395.
Row 8: a(n) = 1920*n^7 + 11200*n^6 + 187488*n^5 + 274480*n^4 + 1593960*n^3 + 286900*n^2 + 972162*n - 135135.
Row 9: a(n) = 4352*n^8 + 34816*n^7 + 761600*n^6 + 1784320*n^5 + 13962848*n^4 + 7874944*n^3 + 29641200*n^2 - 2080800*n + 2027025.
Row 10: a(n) = 9728*n^9 + 102144*n^8 + 2830848*n^7 + 9645312*n^6 + 98382912*n^5 + 106720416*n^4 + 522283552*n^3 + 69265488*n^2 + 255468870*n - 34459425.

Crossrefs

A167572 is the ED3 array.
A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the ED3 array.
A014480, A167581, A167582, A168305 and A168306 equal the first five left hand triangle columns.
A001147 equals the first right hand triangle column.
A167576 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Extensions

Comment and links added by Johannes W. Meijer, Nov 23 2009

A167552 A triangle related to the a(n) formulas of the rows of the ED1 array A167546.

Original entry on oeis.org

1, 3, -2, 5, -5, 2, 7, -7, 14, -8, 9, -6, 63, -66, 24, 11, 0, 209, -264, 308, -144, 13, 13, 559, -689, 2236, -2132, 720, 15, 35, 1281, -1255, 11640, -14980, 14064, -5760, 17, 68, 2618, -1360, 47753, -68068, 145452, -126480, 40320
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009, Nov 23 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED1 array A167546.
The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A003148 and A007318.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 3*n - 2.
Row 3: a(n) = 5*n^2 - 5*n + 2.
Row 4: a(n) = 7*n^3 - 7*n^2 + 14*n - 8.
Row 5: a(n) = 9*n^4 - 6*n^3 + 63*n^2 - 66*n + 24.
Row 6: a(n) = 11*n^5 + 0*n^4 + 209*n^3 - 264*n^2 + 308*n - 144.
Row 7: a(n) = 13*n^6 +13*n^5 +559*n^4 -689*n^3 +2236*n^2 -2132*n +720.
Row 8: a(n) = 15*n^7 + 35*n^6 + 1281*n^5 - 1255*n^4 + 11640*n^3 - 14980*n^2 + 14064*n - 5760.
Row 9: a(n) = 17*n^8 + 68*n^7 + 2618*n^6 - 1360*n^5 + 47753*n^4 - 68068*n^3 + 145452*n^2 - 126480*n + 40320.
Row 10: a(n) = 19*n^9 + 114*n^8 + 4902*n^7 + 684*n^6 + 163419*n^5 - 224694*n^4 + 1048268*n^3 - 1308264*n^2 + 1081632*n - 403200.

Crossrefs

A167546 is the ED1 array.
A000012, A016777, 2*A005891, A167547, A167548 and A167549 are the first sixth ED1 array rows.
A098557 and A167553 equal the first two right hand columns of this triangle.
A005408, A167554 and A167555, A168302 and A168303 equal the first five left hand columns of this triangle.
A000142 equals the row sums.
Cf. A003148 and A007318.

A167565 A triangle related to the a(n) formulas for the rows of the ED2 array A167560.

Original entry on oeis.org

1, 2, 0, 3, 1, 2, 4, 4, 16, 0, 5, 10, 67, 14, 24, 6, 20, 202, 124, 368, 0, 7, 35, 497, 601, 2736, 444, 720, 8, 56, 1064, 2120, 13712, 6464, 16896, 0, 9, 84, 2058, 6096, 53121, 48876, 186732, 25584, 40320, 10, 120, 3684, 15168, 171258, 257640, 1350296
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED2 array A167560.
The recurrence relations for the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A003148 and A007318.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 2*n + 0.
Row 3: a(n) = 3*n^2 + 1*n + 2.
Row 4: a(n) = 4*n^3 + 4*n^2 + 16*n + 0.
Row 5: a(n) = 5*n^4 + 10*n^3 + 67*n^2 + 14*n + 24.
Row 6: a(n) = 6*n^5 + 20*n^4 + 202*n^3 + 124*n^2 + 368*n + 0.
Row 7: a(n) = 7*n^6 + 35*n^5 + 497*n^4 + 601*n^3 + 2736*n^2 + 444*n + 720.
Row 8: a(n) = 8*n^7 + 56*n^6 + 1064*n^5 + 2120*n^4 + 13712*n^3 + 6464*n^2 + 16896*n + 0.
Row 9: a(n) = 9*n^8 + 84*n^7 + 2058*n^6 + 6096*n^5 + 53121*n^4 + 48876*n^3 + 186732*n^2 + 25584*n + 40320.
Row 10: a(n) = 10*n^9 + 120*n^8 + 3684*n^7 + 15168*n^6 + 171258*n^5 + 257640*n^4 + 1350296*n^3 + 533472*n^2 + 1297152*n + 0.

Crossrefs

A167560 is the ED2 array.
A000012, A005843 (n=>1), 2*A104249 (n=>1), A167561, A167562 and A167563 equal the first sixth rows of the array.
A005359 equals the first right hand triangle column.
A000027, A000292, A167566, A167567 and A168304 equal the first five left hand triangle columns.
A000142 equals the row sums.
Cf. A003148 and A007318.

Extensions

Comment and links added by Johannes W. Meijer, Nov 23 2009

A123746 Numerators of partial sums of a series for 1/sqrt(2).

Original entry on oeis.org

1, 1, 7, 9, 107, 151, 835, 1241, 26291, 40427, 207897, 327615, 3296959, 5293843, 26189947, 42685049, 1666461763, 2749521971, 13266871709, 22115585443, 211386315749, 355490397193, 1684973959237, 2855358497999, 53747636888759
Offset: 0

Views

Author

Wolfdieter Lang, Nov 10 2006

Keywords

Comments

Denominators are given by A046161(n),n>=0.
The alternating sum over central binomial coefficients scaled by powers of 4, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/4^k, has the limit s = lim_{n->infinity} r(n) = 1/sqrt(2). From the expansion of 1/sqrt(1-x) for |x|<1 which extends to x=-1 due to Abel's limit theorem and the convergence of the series s. See the W. Lang link.
(2^n)*n!*r(n) = A003148(n). - Wolfdieter Lang, Oct 06 2008

Examples

			a(3)=9 because r(n)=1-1/2+3/8-5/16 = 9/16 = a(3)/A046161(3).

Links

Crossrefs

Cf. A120088/(2*A120777) partial sums for a series of sqrt(2).
Equals A003148 divided by A049606. - Johannes W. Meijer, Nov 23 2009

Programs

  • GAP
    List([0..30], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/(-4)^k )) ); # G. C. Greubel, Aug 10 2019
  • Magma
    [Numerator( (&+[Binomial(2*k,k)/(-4)^k: k in [0..n]])): n in [0..30]]; // G. C. Greubel, Aug 10 2019
  • Maple
    A123746:=n-> numer(add(binomial(2*k,k)/(-4)^k, k=0..n)); seq(A123746(n), n=0..30); # G. C. Greubel, Aug 10 2019
a := n -> numer(add(binomial(-1/2, j), j=0..n));
seq(a(n), n=0..24); # Peter Luschny, Sep 26 2019
  • Mathematica
    Table[Numerator[Sum[Binomial[2*k, k]/(-4)^k, {k,0,n}]], {n,0,30}] (* G. C. Greubel, Mar 28 2018 *)
  • PARI
    {r(n) = sum(k=0,n,(-1/4)^k*binomial(2*k,k))};
vector(30, n, n--; numerator(r(n)) ) \\ G. C. Greubel, Mar 28 2018
  • Sage
    [numerator( sum(binomial(2*k,k)/(-4)^k for k in (0..n)) ) for n in (0..30)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k* binomial(2*k,k)/4^k, n>=0.
r(n) = Sum_{k=0..n} (-1)^k*(2*k-1)!!/(2*k)!!, n>=0, with the double factorials A001147 and A000165.
r(n) = 1/sqrt(2) - binomial(-1/2, 1 + n)*hypergeom([1, 3/2 + n], [2 + n], -1). - Peter Luschny, Sep 26 2019

A091520 Expansion of 1 / ((1 - 4*x) * sqrt(1 + 4*x)) in powers of x.

Original entry on oeis.org

1, 2, 14, 36, 214, 604, 3340, 9928, 52582, 161708, 831588, 2620920, 13187836, 42350744, 209519576, 682960784, 3332923526, 10998087884, 53067486836, 176924683544, 845545262996, 2843923177544, 13479791673896, 45685735967984
Offset: 0

Views

Author

Michael Somos, Jan 18 2004

Keywords

Examples

			G.f. = 1 + 2*x + 14*x^2 + 36*x^3 + 214*x^4 + 604*x^5 + 3340*x^6 + 9928*x^7 + ...

Links

Programs

  • Mathematica
    CoefficientList[Series[1/((1-4x)Sqrt[1+4x]),{x,0,30}],x] (* Harvey P. Dale, Oct 14 2013 *)
Table[2^n (2n+1)!! Hypergeometric2F1[-n, 1/2, 3/2, 2]/n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
RecurrenceTable[{a[n+1] == 4*a[n] - (-1)^(n)*Binomial[2n+2,n+1], a[0]==1}, a, {n,0, 30}] (* G. C. Greubel, Nov 02 2015 *)
  • PARI
    {a(n) = if( n<0, 0, 4^n * sum( k=0, n, binomial(2*k, k) / (-4)^k))};
  • PARI
    my(x='x+O('x^50)); Vec(1/((1-4*x)*sqrt(1+4*x))) \\ Altug Alkan, Nov 02 2015

Formula

G.f.: 1 / ((1 - 4*x) * sqrt(1 + 4*x)).
D-finite with recurrence: n*a(n) = 2 * a(n-1) + 8 * (2*n - 1) * a(n-2).
a(n) = 4^n * Sum_{k=0, n} binomial( 2*k, k) / (-4)^k.
a(n) = A003148(n) * 2^n / n!. - Michael Somos, Mar 17 2011
Asymptotics: a(n) ~ 4^n / sqrt(2).
G.f.: y = A(x) satisfies 0 = (16*x^2 - 1) * y' + (24*x + 2) * y and 0 = y'^3 + 8 * y'^2 * y^3 + 216 * y^5 - 256 * y^7.
G.f.: 1/((1-4*x)*sqrt(1+4*x)) = 1/(1-4*x+2*x*(1-4*x)/G(0)) ; G(k) = 1 + x/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2011
From Vladimir Reshetnikov, Nov 01 2015: (Start)
a(n) = 2^(n-1)*(sqrt(2)*2^n + (-1)^n*(2*n+1)!!*hypergeom([1,n+3/2], [n+2], -1)/(n+1)!).
a(n) = 2^n*(2*n+1)!!*hypergeom([-n,1/2], [3/2], 2)/n!. (End)
a(n+1) = 4*a(n) - (-1)^n*binomial(2*n+2,n+1). - G. C. Greubel, Nov 02 2015
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