A051617 -id:A051617 - cp's OEIS frontend

A051618 a(n) = (4*n+6)(!^4)/6(!^4).

1, 10, 140, 2520, 55440, 1441440, 43243200, 1470268800, 55870214400, 2346549004800, 107941254220800, 5397062711040000, 291441386396160000, 16903600410977280000, 1048023225480591360000, 69169532881719029760000, 4841867301720332083200000, 358298180327304574156800000

Offset: 0

Wolfdieter Lang

This sequence is related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).
Row m=6 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.
a(n) = A001813 a(n+2)/12. - Zerinvary Lajos, Feb 15 2008
For n>4, a(n) mod n^2 = n*(n-2) if n is prime, otherwise 0. - Gary Detlefs, Apr 16 2012

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

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

[Factorial(2*n+4)/(12*Factorial(n+2)): n in [0..100]]; // Vincenzo Librandi, Jul 04 2015

seq(mul((n+2+k), k=1..n+2)/12, n=0..17); # Zerinvary Lajos, Feb 15 2008
A051618 := n -> 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3:
seq(simplify(A051618(n)), n = 0..19);  # Peter Luschny, Jan 22 2025

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 9, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
f[n_] := (2n + 4)!/(12(n + 2)!); Array[f, 16, 0] (* Or *)
FoldList[ #2*#1 &, 1, Range[10, 66, 4]] (* Robert G. Wilson v *)
With[{nn=20},CoefficientList[Series[1/(1-4x)^(5/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 24 2015 *)
Table[(Product[(4*k + 6), {k, 0, n}])/6, {n, 0, 50}] (* G. C. Greubel, Jan 27 2017 *)

A051618(n):=(2*n+4)!/(12*(n+2)!)$
makelist(A051618(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

for(n=0,25, print1((2*n+3)!/(6*(n+1)!), ", ")) \\ G. C. Greubel, Jan 27 2017

a(n) = ((4*n+6)(!^4))/6(!^4).
E.g.f.: 1/(1-4*x)^(5/2).
a(n) = (2n+4)!/(12(n+2)!). -  Gary Detlefs, Mar 06 2011
a(n) = (2*n+3)!/(6*(n+1)!). - Gary Detlefs, Apr 16 2012
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) = (4^(1+n)*Gamma(5/2+n))/(3*sqrt(Pi)). - Gerry Martens, Jul 02 2015
a(n) ~ 2^(2*n+5/2) * n^(n+2) / (3*exp(n)). - Vaclav Kotesovec, Jul 04 2015
a(n) = 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3. - Peter Luschny, Jan 22 2025

A051622 a(n) = (4n+10)(!^4)/10(!^4), related to A000407 ((4n+2)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 14, 252, 5544, 144144, 4324320, 147026880, 5587021440, 234654900480, 10794125422080, 539706271104000, 29144138639616000, 1690360041097728000, 104802322548059136000, 6916953288171902976000, 484186730172033208320000

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.
From Zerinvary Lajos, Feb 15 2008: (Start)
a(n) = A001813(n+3)/120.
a(n) = A051618(n+1)/10. (End)

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-A051621 (rows m=0..9).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(14/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

seq(mul((n+3+k), k=1..n+3)/120, n=0..18); # Zerinvary Lajos, Feb 15 2008

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(7/2), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(14/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+10)(!^4))/10(!^4) = A000407(n+2)/(6*10).
E.g.f.: 1/(1-4*x)^(7/2).
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+7)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013

A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

Original entry on oeis.org

1, 1, 12, 231, 6144, 208845, 8648640, 422463195, 23781703680, 1515973484025, 107941254220800, 8491022274509775, 731304510986649600, 68444451854354701125, 6916953288171902976000, 750681472158682148959875, 87076954662428278259712000, 10751175443940144673035200625

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*6 = 12;
a(3) = 3*7*11 = 231;
a(4) = 4*8*12*16 = 6144;
a(5) = 5*9*13*17*21 = 208845, etc.

Index entries for sequences related to factorial numbers

Column k=4 of A303489.

Cf. A000407, A001813, A007696, A008545, A034176, A034177, A047053, A051617, A051618, A051619, A051620, A051621, A051622, A113551, A303486, A303488.

Table[n! SeriesCoefficient[1/(1 - 4 x)^(n/4), {x, 0, n}], {n, 0, 17}]
Table[Product[4 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[4^n Pochhammer[n/4, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (4*k + n).
a(n) = 4^n*Gamma(5*n/4)/Gamma(n/4).
a(n) ~ 5^(5*n/4-1/2)*n^n/exp(n).

A051620 a(n) = (4n+8)(!^4)/8(!^4), related to A034177(n+1) ((4n+4)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 12, 192, 3840, 92160, 2580480, 82575360, 2972712960, 118908518400, 5231974809600, 251134790860800, 13059009124761600, 731304510986649600, 43878270659198976000, 2808209322188734464000, 190958233908833943552000

Offset: 0

Wolfdieter Lang

Row m=8 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)^(12/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

G(x):=(1-4*x)^(n-4): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=0..15); # Zerinvary Lajos, Apr 04 2009

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn=20},CoefficientList[Series[1/(1-4*x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 10 2017 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(12/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+8)(!^4))/8(!^4) = A034177(n+2)/8.
E.g.f.: 1/(1-4*x)^3.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+6)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013

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

Original entry on oeis.org

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.

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

cp's OEIS Frontend

A051618 a(n) = (4*n+6)(!^4)/6(!^4).

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A051622 a(n) = (4*n+10)(!^4)/10(!^4), related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).

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A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

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A051620 a(n) = (4*n+8)(!^4)/8(!^4), related to A034177(n+1) ((4*n+4)(!^4) quartic, or 4-factorials).

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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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A051622 a(n) = (4n+10)(!^4)/10(!^4), related to A000407 ((4n+2)(!^4) quartic, or 4-factorials).

A051620 a(n) = (4n+8)(!^4)/8(!^4), related to A034177(n+1) ((4n+4)(!^4) quartic, or 4-factorials).

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).