A034177 -id:A034177 - cp's OEIS frontend

A051619 a(n) = (4n+7)(!^4)/7(!^4), related to A034176(n+1) ((4n+3)(!^4) quartic, or 4-factorials).

1, 11, 165, 3135, 72105, 1946835, 60351885, 2112315975, 82380323025, 3542353890075, 166490632833525, 8491022274509775, 467006225098037625, 27553367280784219875, 1735862138689405852125, 116302763292190192092375

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(11/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 10, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(11/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(11/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+7)(!^4))/7(!^4) = A034176(n+2)/7.

E.g.f.: 1/(1-4*x)^(11/4).

A051621 a(n) = (4n+9)(!^4)/9(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 13, 221, 4641, 116025, 3364725, 111035925, 4108329225, 168441498225, 7579867420125, 371413503586125, 19684915690064625, 1122040194333683625, 68444451854354701125, 4448889370533055573125, 306973366566780834545625

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(13/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(13/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(13/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+9)(!^4))/9(!^4) = A007696(n+3)/(5*9).

E.g.f.: 1/(1-4*x)^(13/4).

A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 45, 12, 3, 1, 0, 585, 120, 21, 4, 1, 0, 9945, 1680, 231, 32, 5, 1, 0, 208845, 30240, 3465, 384, 45, 6, 1, 0, 5221125, 665280, 65835, 6144, 585, 60, 7, 1, 0, 151412625, 17297280, 1514205, 122880, 9945, 840, 77, 8, 1

Offset: 0

Peter Luschny, Mar 06 2024

The sequence of square arrays A(m, n, k) starts: A094587 (m = 1), A370419 (m = 2), A371077(m = 3), this array (m = 4).

			The array starts:
[0] 1,    1,     1,     1,      1,      1,      1,      1,      1, ...
[1] 0,    1,     2,     3,      4,      5,      6,      7,      8, ...
[2] 0,    5,    12,    21,     32,     45,     60,     77,     96, ...
[3] 0,   45,   120,   231,    384,    585,    840,   1155,   1536, ...
[4] 0,  585,  1680,  3465,   6144,   9945,  15120,  21945,  30720, ...
[5] 0, 9945, 30240, 65835, 122880, 208845, 332640, 504735, 737280, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,      1;
[2] 0,      1,     1;
[3] 0,      5,     2,    1;
[4] 0,     45,    12,    3,   1;
[5] 0,    585,   120,   21,   4,  1;
[6] 0,   9945,  1680,  231,  32,  5, 1;
[7] 0, 208845, 30240, 3465, 384, 45, 6, 1;

Similar square arrays: A094587, A370419, A371077.
Rows: A000012, A001477, A028347, A370914.
Columns: A000007, A007696, A001813, A008545, A047053, A007696, A000407, A034176, A052570 and A034177, A051617, A051618, A051619, A051620.
Cf. A370913 (row sums of triangle), A371026.

A := (n, k) -> 4^n*pochhammer(k/4, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 4*x)^(-k/4);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..5);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-4)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371026(n-1, k-1)):
U := Matrix(7, 7, (n, k) -> binomial(n-1, k-1)):
MatrixMatrixMultiply(L, Transpose(U));

A[n_, k_] := 4^n * Pochhammer[k/4, n]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)

def A(n, k): return 4**n * rising_factorial(k/4, n)
for n in range(6): print([A(n, k) for k in range(9)])

A(n, k) = 4^n*Product_{j=0..n-1} (j + k/4).
A(n, k) = 4^n*Gamma(k/4 + n) / Gamma(k/4) for k >= 1.
The exponential generating function for column k is (1 - 4*x)^(-k/4). But much more is true: (1 - m*x)^(-k/m) are the exponential generating functions for the columns of the arrays A(m, n, k) = m^n*Pochhammer(k/m, n).
The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-4)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
In A370419 Werner Schulte pointed out how A371025 is related to the LU decomposition of A370419. Here the same procedure can be used and amounts to A = A371026 * transpose(binomial triangle), where '*' denotes matrix multiplication. See the Maple section for an implementation.

A052570 E.g.f.: x/(1-4*x).

Original entry on oeis.org

0, 1, 8, 96, 1536, 30720, 737280, 20643840, 660602880, 23781703680, 951268147200, 41855798476800, 2009078326886400, 104472072998092800, 5850436087893196800, 351026165273591808000, 22465674577509875712000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 513

A034177 is an essentially identical sequence. - Philippe Deléham, Sep 18 2008

spec := [S,{S=Prod(Z,Sequence(Union(Z,Z,Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[x/(1-4x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 15 2017 *)

Recurrence: {a(1)=1, a(0)=0, (-4-4*n)*a(n)+a(n+1)=0.}

4^(n-1)*n!=n!*A000302(n-1). for n >= 1.

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

A098560 Expansion of e.g.f. (1+4x)/(1-4x).

Original entry on oeis.org

1, 8, 64, 768, 12288, 245760, 5898240, 165150720, 5284823040, 190253629440, 7610145177600, 334846387814400, 16072626615091200, 835776583984742400, 46803488703145574400, 2808209322188734464000, 179725396620079005696000

Offset: 0

Paul Barry, Sep 14 2004

G. C. Greubel, Table of n, a(n) for n = 0..360

Cf. A098559, A098558, A032184.

[1] cat [2^(2*n+1)*Factorial(n): n in [2..30]]; // G. C. Greubel, Jan 17 2018

s=2;lst={1};Do[s+=n*s+s;AppendTo[lst, s], {n, 2, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn=20},CoefficientList[Series[(1+4x)/(1-4x),{x,0,nn}],x] Range[0,nn]!] (* or *) Join[{1},Table[2*4^n n!,{n,20}]] (* Harvey P. Dale, Jan 16 2012 *)

for(n=0, 30, print1(if(n==0,1, 2^(2*n+1)*n!), ", ")) \\ G. C. Greubel, Jan 17 2018

a(n) = 2*4^n*n! - 0^n.
a(n+1) = 8*A034177(n).
a(n) - 4*n*a(n-1) = 0. - R. J. Mathar, Dec 21 2014
Sum_{n>=0} 1/a(n) = (exp(1/4)+1)/2. - Amiram Eldar, Feb 02 2023

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

A051619 a(n) = (4*n+7)(!^4)/7(!^4), related to A034176(n+1) ((4*n+3)(!^4) quartic, or 4-factorials).

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A051621 a(n) = (4*n+9)(!^4)/9(!^4), related to A007696(n+1) ((4*n+1)(!^4) quartic, or 4-factorials).

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A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

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A052570 E.g.f.: x/(1-4*x).

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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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A098560 Expansion of e.g.f. (1+4*x)/(1-4*x).

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

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A051619 a(n) = (4n+7)(!^4)/7(!^4), related to A034176(n+1) ((4n+3)(!^4) quartic, or 4-factorials).

A051621 a(n) = (4n+9)(!^4)/9(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

A098560 Expansion of e.g.f. (1+4x)/(1-4x).