A081405 -id:A081405 - cp's OEIS frontend

A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).

0, 0, 1, 1, 2, 3, 4, 3, 4, 4, 5, 5, 6, 7, 9, 7, 8, 8, 10, 9, 10, 10, 11, 11, 11, 12, 12, 11, 13, 13, 14, 11, 13, 14, 14, 13, 14, 14, 16, 15, 16, 18, 19, 19, 19, 19, 21, 19, 20, 19, 21, 20, 21, 21, 21, 19, 20, 20, 22, 22, 24, 25, 25, 23, 23, 23, 24, 24, 27, 26, 27, 25, 27, 28, 29, 28

Offset: 0

Labos Elemer, Mar 28 2003

It is easy to show that a(n) is between n/log(n) and 2n/log(n) (for n>n0), cf. [Campbell 1984]. The sequence A137687, roughly the middle of this interval, is a fair approximation for A081399. See A137686 for the (signed) difference of the two sequences.

M. F. Hasler, Table of n, a(n) for n = 0..3000.
Douglas M. Campbell, The Computation of Catalan Numbers, Mathematics Magazine, Vol. 57, No. 4. (Sep., 1984), pp. 195-208.

Cf. A001222, A000108, A022559, A023816, A048621, A081405, A120626, A137686-A137687.

Cf. A080405.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(2*n,n)/(1+n)) fi end: seq(a(n), n=0..75); # Zerinvary Lajos, Apr 11 2008

a[n_] := PrimeOmega[ CatalanNumber[n]]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, Jul 02 2013 *)

A081399(n)=bigomega(prod(i=2,n,(n+i)/i)) \\ M. F. Hasler, Feb 06 2008

a(n)=A001222[A000108(n)]

Edited and extended by M. F. Hasler, Feb 06 2008

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

A249159 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 4, 7, 2, 2, 15, 18, 24, 4, 4, 24, 57, 30, 36, 4, 4, 105, 174, 282, 88, 100, 8, 8, 192, 561, 414, 570, 120, 132, 8, 8, 945, 1950, 3660, 1620, 2040, 312, 336, 16, 16, 1920, 6555, 6090, 9360, 2820, 3360, 392, 416, 16, 16, 10395, 25290, 53370

Offset: 0

Clark Kimberling, Oct 23 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = 1 + n)/(2*f(n-1,x)), where f(0,x) = 1.
(Sum of numbers in row n) = A000982(n+1) for n >= 0.
Column 1 is essentially A081405.

Clark Kimberling, Rows 0..100, flattened

Cf. A000982, A081405.

z = 15; f[x_, n_] := 1 + n/(2 f[x, n - 1]); f[x_, 1] = 1;
t = Table[Factor[f[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249159 array *)
Flatten[CoefficientList[u, x]] (* A249159 sequence *)

f(0,x) = 1/1, so that p(0,x) = 1
f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
f(2,x) = (3 + 2 x + x^2)/(1 + x), so that p(2,x) = 3 + 2 x + x^2.
First 6 rows of the triangle of coefficients:
1
1    1
3    2     2
4    7     2     2
15   18    24    4     4
24   57    30    36    4    4

A081406 a(n) = (n+1)*a(n-3), a(0)=a(1)=a(2)=1 for n>1.

Original entry on oeis.org

1, 1, 1, 4, 5, 6, 28, 40, 54, 280, 440, 648, 3640, 6160, 9720, 58240, 104720, 174960, 1106560, 2094400, 3674160, 24344320, 48171200, 88179840, 608608000, 1252451200, 2380855680, 17041024000, 36321084800, 71425670400, 528271744000, 1162274713600

Offset: 0

Labos Elemer, Apr 01 2003

nonn

			a(3n+2)=A034001[n]; while other subsequences are near(but not equal) to A001669, A000359.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A002866, A000142, A001147, A081405.

a:= function(k)
    if k<3 then return 1;
    elif k<6 then return k+1;
    else return (k+1)*a(k-3);
    fi;
  end;
List([0..35], n-> a(n) ); # G. C. Greubel, Aug 24 2019

a:= func< n | n le 2 select 1 else n in [3..5] select n+1 else (n+1)*Self(n-2) >;
[a(n): n in [0..35]]; // G. C. Greubel, Aug 24 2019

f[n_]:= (n+1)*f[n-3]; f[0]=1; f[1]=1; f[2]=1; Table[f[n], {n, 30}]
RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(n+1)a[n-3]},a,{n,30}] (* Harvey P. Dale, Mar 06 2019 *)

a(n) = if(n<3, 1, (n+1)*a(n-3) );
vector(35, n, a(n-1)) \\ G. C. Greubel, Aug 24 2019

def a(n):
    if n<3: return 1
    elif 3<= n <= 5: return n+1
    else: return (n+1)*a(n-3)
[a(n) for n in (0..35)] # G. C. Greubel, Aug 24 2019

Corrected and extended by Harvey P. Dale, Mar 06 2019

A081408 a(n) = (n+1)*a(n-5), with a(0)=a(1)=a(2)=a(3)=a(4)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 7, 8, 9, 10, 66, 84, 104, 126, 150, 1056, 1428, 1872, 2394, 3000, 22176, 31416, 43056, 57456, 75000, 576576, 848232, 1205568, 1666224, 2250000, 17873856, 27143424, 39783744, 56651616, 78750000, 643458816, 1004306688, 1511782272

Offset: 0

Labos Elemer, Apr 01 2003

nonn

Quintic factorial sequences are generated by single 5-order recursion and appear in unified form.

			A008548, A034323, A034300, A034301, A034325 sequences are combed together as A081408(5n+r) with r=0,1,2,3,4.

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

Cf. A001147, A002866, A034001, A007599, A034000, A007696, A000407, A034176, A034177, A008548, A034323, A034300, A034301, A034325 [double, triple, quartic, quintic, factorial subsequences], generated together in A081405-A081408.

a:=[1,1,1,1,1];; for n in [6..40] do a[n]:=n*a[n-5]; od; a; # G. C. Greubel, Aug 15 2019

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

[n le 5 select 1 else n*Self(n-5): n in [1..40]]; // G. C. Greubel, Aug 15 2019

a[0]=a[1]=a[2]=a[3]=a[4]=1; a[x_]:= (x+1)*a[x-5]; Table[a[n], {n, 40}]

m=30; v=concat([1,1,1,1,1], vector(m-5)); for(n=6, m, v[n]=n*v[n-5] ); v \\ G. C. Greubel, Aug 15 2019

def a(n):
    if (n<5): return 1
    else: return (n+1)*a(n-5)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 15 2019

cp's OEIS Frontend

A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Mathematica

PARI

Sage

A249159 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A081406 a(n) = (n+1)*a(n-3), a(0)=a(1)=a(2)=1 for n>1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Extensions

A081408 a(n) = (n+1)*a(n-5), with a(0)=a(1)=a(2)=a(3)=a(4)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Mathematica

PARI

Sage