A007696 -id:A007696 - cp's OEIS frontend

A235136 a(n) = (2n - 1) a(n-2) for n>1, a(0) = a(1) = 1.

1, 1, 3, 5, 21, 45, 231, 585, 3465, 9945, 65835, 208845, 1514205, 5221125, 40883535, 151412625, 1267389585, 4996616625, 44358635475, 184874815125, 1729986783525, 7579867420125, 74389431691575, 341094033905625, 3496303289504025, 16713607661375625

Offset: 0

Michael Somos, Jan 03 2014

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 21*x^4 + 45*x^5 + 231*x^6 + 585*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..730

Cf. A007662, A007696, A008545, A196265.

Cf. A068465, A068466.

a[ n_] := 2^n If[ OddQ[n], 2 Pochhammer[ 1/4, (n + 1)/2], Pochhammer[ 3/4, n/2]]; (* Michael Somos, Jan 16 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (-2 Gamma[5/2] HermiteH[ -3/2, x] + (3 Gamma[5/4] + 2 Gamma[7/4]) Hypergeometric1F1[ 3/4, 1/2, x^2]) / (3 Gamma[5/4]), {x, 0, n}] // FullSimplify]; (* Michael Somos, Jan 16 2014 *)
RecurrenceTable[{a[0]==a[1]==1, a[n]==(2 n - 1) a[n - 2]}, a, {n, 25}] (* Vincenzo Librandi, Aug 08 2018 *)

{a(n) = if( n<0, (-1)^(-n\2) / a(-1-n), if( n<2, 1, (2*n - 1) * a(n-2)))};

Let b(n) = a(2*n - 2) / a(2*n + 1). Then b(-n) = b(n), 0 = b(n+1) * (b(n+1) + 2*b(n+2)) + b(n) * (2*b(n+1) - 5*b(n+2)) for all n in Z.
a(n-1) + a(n-2) = A196265(n) if n>1.
a(2*n) = A008545(n). a(2*n - 1) = A007696(n). a(n) = A007662(2*n - 1).
E.g.f. A(x) =: y satisfies 0 = y * 3 + y' * 2*x - y''.
0 = a(n)*(2*a(n+1) - a(n+3)) + a(n+1)*(a(n+2)) for all n in Z. - Michael Somos, Jan 24 2014
Let b(n) = a(n - 2) / a(n + 1). Then b(-n) = (-1)^n * b(n), 0 = b(n) * (b(n+1) - 4*b(n+3)) + b(n+2) * (2*b(n+1) + b(n+3)) for all n in Z. - Michael Somos, Sep 13 2014
a(n) ~ c * sqrt(Pi) * (2*n)^(n/2+1/4) / exp(n/2), where c = 2/Gamma(1/4) if n is odd, and 1/Gamma(3/4) if n is even. - Amiram Eldar, Sep 01 2025

A347012 E.g.f.: exp(x) / (1 - 4 * x)^(1/4).

Original entry on oeis.org

1, 2, 8, 64, 800, 13376, 278272, 6914048, 199629824, 6566164480, 242327576576, 9915111636992, 445432721932288, 21795710738038784, 1153805878313615360, 65700181140859518976, 4004182878034473254912, 260071258357260225609728, 17932703649301871611346944

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Binomial transform of A007696.

Cf. A000522, A007696, A056545, A084262, A088991, A346258, A347013, A347014.

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021

nmax = 18; CoefficientList[Series[Exp[x]/(1 - 4 x)^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]
Table[HypergeometricU[1/4, n + 5/4, 1/4]/Sqrt[2], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n,k) * A007696(k).

a(n) ~ n! * exp(1/4) * 4^n / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 14 2021

A371026 Triangle read by rows: T(n, k) = 4^nSum_{j=0..k} (-1)^(k - j)binomial(k, j)* Pochhammer(j/4, n).

Original entry on oeis.org

1, 0, 1, 0, 5, 2, 0, 45, 30, 6, 0, 585, 510, 180, 24, 0, 9945, 10350, 4950, 1200, 120, 0, 208845, 247590, 144900, 48600, 9000, 720, 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040, 0, 151412625, 216093150, 164872260, 80713080, 26334000, 5594400, 705600, 40320

Offset: 0

Peter Luschny, Mar 08 2024

			Triangle read by rows:
  [0] 1;
  [1] 0,       1;
  [2] 0,       5,       2;
  [3] 0,      45,      30,       6;
  [4] 0,     585,     510,     180,      24;
  [5] 0,    9945,   10350,    4950,    1200,    120;
  [6] 0,  208845,  247590,  144900,   48600,   9000,   720;
  [7] 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040;

Cf. A000142 (main diagonal), A007696 (column 1), A371027 (row sums).

Cf. A371025.

A371026 := (n, k) -> local j; 4^n*add((-1)^(k - j)*binomial(k, j)*pochhammer(j/4, n), j = 0..k): seq(seq(A371026(n, k), k = 0..n), n = 0..9);

from functools import cache
@cache
def T(n, k):   # After Werner Schulte
    if k == 0: return 0**n
    if k == n: return n * T(n-1, n-1)
    return k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k)
for n in range(8): print([T(n, k) for k in range(n + 1)])
# Peter Luschny, Mar 17 2024

T(n, k) = k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = n! for n >= 0. - Werner Schulte, Mar 17 2024

A020042 a(n) = round( Gamma(n+1/4)/Gamma(1/4) ).

Original entry on oeis.org

1, 0, 0, 1, 2, 10, 51, 319, 2310, 19061, 176310, 1807181, 20330789, 249052162, 3299941145, 47024161317, 717118460090, 11653174976468, 201017268344078, 3668565147279424, 70619879085128903, 1430052551473860293

Offset: 0

Simon Plouffe

nonn

Gamma(n+1/4)/Gamma(1/4) = 1, 1/4, 5/16, 45/64, 585/256, 9945/1024, 208845/4096, 5221125/16384, 151412625/65536, ... - R. J. Mathar, Sep 04 2016

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A007696, A020087, A020132.

[Round(Gamma(n+1/4)/Gamma(1/4)): n in [0..30]]; // G. C. Greubel, Dec 06 2019

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;
seq( round(pochhammer(1/4,n)), n=0..30); # G. C. Greubel, Dec 06 2019

Table[Round[Pochhammer[1/4,n]], {n,0,30}] (* G. C. Greubel, Dec 06 2019 *)

x=1/4; vector(30, n, round(gamma(n-1+x)/gamma(x)) ) \\ G. C. Greubel, Dec 06 2019

[round(rising_factorial(1/4,n)) for n in (0..30)] # G. C. Greubel, Dec 06 2019

A024383 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) = 4*k - 3 for k = 1, 2, 3, ....

Original entry on oeis.org

1, 4, 41, 488, 8881, 176556, 4622745, 128838480, 4403082465, 157917434580, 6659489632905, 292097166060600, 14653855170875025, 759940716395000700, 44202442040567948025, 2645857155729629066400, 175060715455871850866625

Offset: 1

Clark Kimberling

Cf. A024199, A024217, A024396, A007696, A181048.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 4 else 4*a(n-1) + (4*n - 3)^2*a(n-2) end if; end:
seq(a(n), n = 0..20);

Table[Product[4*k - 3, {k, 1, n}] * Sum[(-1)^(k+1)/(4*k - 3), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jan 02 2020 *)

a(n) = prod(k=1, n, 4*k-3)*sum(k=1, n, (-1)^(k+1)/(4*k-3)); \\ Michel Marcus, Jul 06 2019

a(n) ~ (Pi^(3/2) + 2*sqrt(Pi)*log(1 + sqrt(2))) * 2^(2*n - 2) * n^(n - 1/4) / (Gamma(1/4) * exp(n)). - Vaclav Kotesovec, Jan 02 2020
From Peter Bala, Mar 21 2024: (Start)
a(n) = Product_{k = 0..n} (4*k + 1) * Sum_{k = 0..n} (-1)^k/(4*k + 1).
a(n) = 4*a(n-1) + (4*n - 3)^2*a(n-2) with a(0) = 1 and a(1) = 4.
b(n) := Product_{k = 0..n} (4*k + 1) = A007696(n+1) satisfies the same 3-term recurrence with b(0) = 1 and b(1) = 5, leading to the continued fraction expansion for the constant A181048 = Sum_{k >= 0} (-1)^k/(4*k + 1) = 1/(1 + 1^2/(4 + 5^2/(4 + 9^2/(4 + 13^2/(4 + ... ))))) due to Euler. (End)

More terms from Sean A. Irvine, Jul 06 2019

A081407 4th-order non-linear ("factorial") recursion: a(0)=a(1)=a(2)=a(3)=1, a(n) = (n+1)*a(n-4).

Original entry on oeis.org

1, 1, 1, 1, 5, 6, 7, 8, 45, 60, 77, 96, 585, 840, 1155, 1536, 9945, 15120, 21945, 30720, 208845, 332640, 504735, 737280, 5221125, 8648640, 13627845, 20643840, 151412625, 259459200, 422463195, 660602880, 4996616625, 8821612800

Offset: 0

Labos Elemer, Apr 01 2003

nonn

			Following sequences are interleaved: A007696: {5,45,585,..}; A000404: {6,60,840,..} A034176: {7,77,1155,..}; A034177: {8,96,1536,..}

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000404, A007696, A034176, A034177, A081405, A081405, A081407.

a:= function(k)
    if k<4 then return 1;
    elif k<7 then return k+1;
    else return (k+1)*a(k-4);
    fi;
  end;
List([0..35], n-> a(n) ); # G. C. Greubel, Aug 24 2019

a081407 n = a081408_list !! n
a081407_list = 1 : 1 : 1 : 1 : zipWith (*) [5..] a081407_list
-- Reinhard Zumkeller, Jan 05 2012

a:= func< n | n le 3 select 1 else n in [4..6] select n+1 else (n+1)*Self(n-3) >;
[a(n): n in [0..35]]; // G. C. Greubel, Aug 24 2019

f[n_]:= (n+1)*f[n-4]; f[0]=1; f[1]=1; f[2]=1; f[3]=1; Table[f[n], {n, 0, 40}]
nxt[{n_,a_,b_,c_,d_}]:={n+1,b,c,d,a(n+2)}; NestList[nxt,{3,1,1,1,1},40][[;;,2]] (* Harvey P. Dale, Jan 13 2025 *)

a(n) = if(n<4, 1, (n+1)*a(n-4) );
vector(35, n, a(n-1)) \\ G. C. Greubel, Aug 24 2019

def a(n):
    if n<4: return 1
    elif 4<= n <= 6: return n+1
    else: return (n+1)*a(n-4)
[a(n) for n in (0..35)] # G. C. Greubel, Aug 24 2019

A091544 First column sequence of array A091746 ((6,2)-Stirling2).

Original entry on oeis.org

1, 30, 2700, 491400, 150368400, 69470200800, 45155630520000, 39285398552400000, 44078217175792800000, 61973973349164676800000, 106719182107261573449600000, 220908706962031457040672000000, 541226332056977069749646400000000, 1548989762347068373623487996800000000

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also fifth column (m=4) sequence of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A091535 (third column of A091543, first column of array A091534), A000407, A007696, A091746.

a[n_] := 2^(4*n-1) * Pochhammer[1/4, n] * Pochhammer[1/2, n]; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

a(n) = 2^(n-1)*Product_{j=0..n-1}((2*j+1)*(4*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=1.
a(n) = (2^(4*n-1))*risefac(1/4, n)*risefac(1/2, n), n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac4(4*n-3)*fac4(4*n-2)/2, n>=1, with fac4(4*n-3) = A007696(n) and fac4(4*n-2)/2 = A000407(n+1) (quartic- or 4-factorials).
E.g.f.: (hypergeom([1/4, 1/2], [], 16*x)-1)/2.
a(n) = A091746(n, 2), n>=1.
a(n) ~ sqrt(Pi) * 2^(4*n) * n^(2*n-1/4) / (Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025

A123145 a(1) = 1, a(n) = a(n-1) if n == 1 (mod 4), otherwise a(n) = n * a(n-1) for n >= 2.

Original entry on oeis.org

1, 2, 6, 24, 24, 144, 1008, 8064, 8064, 80640, 887040, 10644480, 10644480, 149022720, 2235340800, 35765452800, 35765452800, 643778150400, 12231784857600, 244635697152000, 244635697152000, 5381985337344000, 123785662758912000, 2970855906213888000

Offset: 1

Roger L. Bagula, Oct 01 2006

nonn

Terms which repeat appear to be denominators of g.f. 0F2(--; 1/2, 3/4; z^4/64), which begin 24, 8064, 10644480, 35765452800, ... - Benedict W. J. Irwin, Jun 15 2018

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A007559, A007696.

function a(n) // a = A123145
  if n eq 1 then return 1;
  elif (n mod 4) eq 1 then return a(n-1);
  else return n*a(n-1);
  end if;
end function;
[a(n): n in [1..40]]; // G. C. Greubel, Jul 16 2023

a:= proc(n) option remember; `if`(n=0, 1,
      `if`(irem(n, 4)=1, 1, n)*a(n-1))
    end:
seq(a(n), n=1..24);  # Alois P. Heinz, Jul 16 2023

a[n_]:= a[n]= If[n==1, 1, If[Mod[n,4]==1, a[n-1], n*a[n-1]]];
Table[a[n], {n,30}]

def a(n): # A123145
    if (n==1): return 1
    elif (n%4==1): return a(n-1)
    else: return n*a(n-1)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 16 2023

Conjecture: E.g.f.: E(x)=d(G(0))/dx where G(k) = 1 + x/(4*k+1 - x*(4*k+1)/(1 + x - x/(1 + x - x/(x + 1/G(k+1) )))), or shift on 1 left G(0); (continued fraction,5-step). - Sergei N. Gladkovskii, Nov 26 2012

Edited by N. J. A. Sloane, Oct 04 2006

More terms from Bruno Berselli, Nov 27 2012

A153274 Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

Original entry on oeis.org

2, 6, 15, 24, 105, 280, 120, 945, 3640, 9945, 720, 10395, 58240, 208845, 576576, 5040, 135135, 1106560, 5221125, 17873856, 49579075, 40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000, 362880, 34459425, 608608000, 4996616625, 26381811456, 104463111025, 337767408000, 939536222625

Offset: 1

Roger L. Bagula, Dec 22 2008

A Pochhammer function-based triangular sequence.

Row sums are: {2, 21, 409, 14650, 854776, 73920791, 8878927331, 1413788600036, 288152651134776, 73152069870215127, ...}.

			Triangle begins as:
      2;
      6,      15;
     24,     105,      280;
    120,     945,     3640,      9945;
    720,   10395,    58240,    208845,    576576;
   5040,  135135,  1106560,   5221125,  17873856,   49579075;
  40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755, A045756.

Cf. A142589, A256268.

Flat(List([1..12], n-> List([1..n], k-> Product([0..n], j-> j*k+1 )))); # G. C. Greubel, Mar 05 2020

[(&*[j*k+1: j in [0..n]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 05 2020

seq(seq( k^(n+1)*pochhammer(1/k, n+1), k=1..n), n=1..12); # G. C. Greubel, Mar 05 2020

Table[Apply[Plus, CoefficientList[j*k^n*Pochhammer[(j+k)/k, n], j]], {n, 12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 05 2020 *)
Table[k^(n+1)*Pochhammer[1/k, n+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 05 2020 *)

T(n, k) = prod(j=0, n, j*k+1);
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Mar 05 2020

[[k^(n+1)*rising_factorial(1/k,n+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 05 2020

T(n, k) = k^(n+1) * Pochmammer(1/k, n+1).

T(n, k) = Product_{j=0..n} (j*k + 1). - G. C. Greubel, Mar 05 2020

Edited by G. C. Greubel, Mar 05 2020

A254796 Denominators of the convergents of the generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).

Original entry on oeis.org

1, 4, 25, 200, 2025, 24300, 342225, 5475600, 98903025, 1978060500, 43616234025, 1046789616600, 27260146265625, 763284095437500, 22925783009390625, 733625056300500000, 24966177697226390625, 898782397100150062500, 34178697267502928765625

Offset: 0

Peter Bala, Feb 23 2015

The generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) represents the constant L^2/Pi = 2.188439... = A254794, where L is the lemniscate constant A062539. See A254795 for the numerators of the convergents of the continued fraction.

			54/25 = 2.16, 441/200 = 2.205 etc approach 2.188..

T. J. Osler, The missing fractions in Brouncker’s sequence of continued fractions for Pi

Cf. A007696, A024199, A062539, A254794, A254795.

I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)+(2*n-3)^2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 24 2015

a[0] := 1: a[1] := 4:
for n from 2 to 18 do a[n] := 4*a[n-1] + (2*n-1)^2*a[n-2] end do:
seq(a[n], n = 0 .. 18);

RecurrenceTable[{a[0] == 1, a[1] == 4, a[n] == 4 a[n - 1] + (2 n - 1)^2 a[n - 2]}, a, {n, 20}] (* Vincenzo Librandi, Feb 24 2015 *)

a(2*n) = A007696(n+1)^2 = ( Product {k = 0..n} 4*k + 1 )^2.
a(2*n-1) = 4*n*A007696(n)^2 = 4*n * ( Product {k = 0..n-1} 4*k + 1 )^2.
a(n) = 4*a(n-1) + (2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 4.
a(2*n+1) = 4*(n + 1)*a(2*n); a(2*n) = (4*n + 2)*a(2*n-1) + a(2*n-2).
Empirical e.g.f.: ((-Q(1/2, -3)-Q(-1/2, -3))*P(1/2, (2*x+3)/(2*x-1))+Q(1/2, (2*x+3)/(2*x-1))*(P(1/2, -3)+P(-1/2, -3)))/((1-2*x)^(3/2)*(-Q(-1/2, -3)*P(1/2, -3)+Q(1/2, -3)*P(-1/2, -3))) where P and Q are Legendre functions of the first and second kinds. - Robert Israel, Feb 24 2015

cp's OEIS Frontend

A235136 a(n) = (2*n - 1) * a(n-2) for n>1, a(0) = a(1) = 1.

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A347012 E.g.f.: exp(x) / (1 - 4 * x)^(1/4).

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A371026 Triangle read by rows: T(n, k) = 4^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j)* Pochhammer(j/4, n).

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A020042 a(n) = round( Gamma(n+1/4)/Gamma(1/4) ).

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Magma

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A024383 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) = 4*k - 3 for k = 1, 2, 3, ....

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A081407 4th-order non-linear ("factorial") recursion: a(0)=a(1)=a(2)=a(3)=1, a(n) = (n+1)*a(n-4).

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A091544 First column sequence of array A091746 ((6,2)-Stirling2).

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A235136 a(n) = (2n - 1) a(n-2) for n>1, a(0) = a(1) = 1.

A371026 Triangle read by rows: T(n, k) = 4^nSum_{j=0..k} (-1)^(k - j)binomial(k, j)* Pochhammer(j/4, n).

A024383 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) = 4*k - 3 for k = 1, 2, 3, ....