A052762 -id:A052762 - cp's OEIS frontend

A001094 a(n) = n + n(n-1)(n-2)*(n-3).

0, 1, 2, 3, 28, 125, 366, 847, 1688, 3033, 5050, 7931, 11892, 17173, 24038, 32775, 43696, 57137, 73458, 93043, 116300, 143661, 175582, 212543, 255048, 303625, 358826, 421227, 491428, 570053, 657750, 755191, 863072, 982113, 1113058

Offset: 0

N. J. A. Sloane, Ray Wills (rwills(AT)vmprofs.estec.esa.nl)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001095, A001096, A052762.

List([0..35], n-> n + 24*Binomial(n,4)); # G. C. Greubel, Aug 26 2019

[n + n*(n-1)*(n-2)*(n-3): n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

seq(n + 4!*binomial(n,4), n=0..35); # G. C. Greubel, Aug 26 2019

Table[n+n(n-1)(n-2)(n-3),{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,1,2,3,28},40] (* Harvey P. Dale, Feb 02 2012 *)

vector(35, n, (n-1) + 4!*binomial(n-1,4)) \\ G. C. Greubel, Aug 26 2019

[n + 24*binomial(n,4) for n in (0..35)] # G. C. Greubel, Aug 26 2019

G.f.: x*(1 -3*x +3*x^2 +23*x^3)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=28. - Harvey P. Dale, Feb 02 2012
From G. C. Greubel, Aug 26 2019: (Start)
a(n) = n + 4!*binomial(n,4).
E.g.f.: x*(1+x^3)*exp(x). (End)

More terms from James Sellers, Sep 19 2000

A190136 Largest prime factor of n(n+1)(n+2)*(n+3).

Original entry on oeis.org

3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 13, 13, 29, 29, 31, 31, 31, 31, 17, 17, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 17, 17, 53, 53, 53, 53, 19, 29, 59, 59, 61, 61, 61, 61, 31, 13, 67, 67, 67

Offset: 1

Reinhard Zumkeller, May 07 2011

nonn

a(n) > 11 for n > 9;
a(A086801(n)) = A000040(n) for n > 2.
It follows from Størmer's theorem that lim inf a(n) = infinity, and in fact a(n) >> log log n. - Charles R Greathouse IV, Feb 19 2013

			Numbers m <= 10^6 such that a(m) = p:
p=13: 10, 11, 12, 13, 24, 25, 63;
p=17: 14, 15, 32, 33, 48, 49;
p=19: 16, 17, 18, 19, 54, 75, 168;
p=23: 20, 21, 22, 23, 207, 322;
p=29: 26, 27, 55, 114;
p=31: 28, 29, 30, 31, 62, 90, 152, 153, 340, 493, 1518;
p=37: 34, 35, 36, 37, 74, 184, 405;
p=41: 38, 39, 123, 245, 285, 286, 287, 492, 1023, 1517, 1680;
p=43: 40, 41, 42, 43, 84, 85, 169, 258, 341, 342, 558, 1330, 1331, 2106, 5289, 10878;
p=47: 44, 45, 46, 47, 91, 92, 93, 185, 186, 187, 374, 375, 702, 986, 987, 17575;
p=53: 50, 51, 52, 53, 159, 368, 369, 527, 845, 899, 900, 1375;
p=59: 56, 57, 115, 116, 117, 118, 174, 294, 528, 529, 530, 648, 943, 1885, 6783;
p=61: 58, 59, 60, 61, 119, 120, 121, 122, 182, 183, 242, 243, 244, 549, 608, 609, 1034, 1218, 1219, 1767, 1768, 2013, 2254, 2622;
p=67: 64, 65, 66, 67, 132, 133, 735, 1271, 1272, 1273, 2208, 2277, 3885, 4958, 5828, 5829;
p=71: 68, 69, 140, 141, 142, 284, 423, 424, 494, 636, 637, 779, 780, 781, 3477, 3478, 3549, 3550, 4899;
p=73: 70, 71, 72, 73, 143, 144, 145, 219, 363, 510, 728, 729, 803, 1022, 1239, 1679, 2772, 70224;
p=79: 76, 77, 78, 79, 158, 234, 235, 472, 473, 474, 550, 867, 868, 1024, 1104, 1419, 2209, 2448, 2923, 3476, 3869, 4898, 5290, 7502, 46136, 70150;
p=83: 80, 81, 82, 83, 246, 247, 413, 495, 663, 664, 1078, 1159, 1824, 2736, 3483, 4232, 4896, 4897, 7137, 8214, 12614, 36517, 97524;
p=89: 86, 87, 88, 89, 175, 264, 265, 354, 531, 710, 711, 712, 798, 1245, 1332, 2847, 4895, 5073, 6318, 18423, 28302, 29279;
p=97: 94, 95, 96, 97, 288, 289, 483, 580, 581, 582, 774, 873, 1064, 1065, 1455, 2132, 2133, 3007, 3975, 4556, 4557, 6496, 6497, 6887, 7564, 7565, 7566, 13869, 17457.

Paulo Ribenboim, Galimatias Arithmeticae (Chap 11), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 345.
J. J. Sylvester, "On arithmetical series", Messenger of Mathematics 21 (1892), pp. 1-19 and 87-120.
M. Faulkner, "On a theorem of Sylvester and Schur", J. London Math. Soc. 41:1 (1966), pp. 107-110.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A074399, A093074, A193945.

a190136 n = maximum $ map a006530 [n..n+3]

Table[FactorInteger[Times@@(n+Range[0,3])][[-1,1]],{n,70}] (* Harvey P. Dale, Mar 19 2018 *)

gpf(n)=vecmax(factor(n)[,1])
a(n)=my(p=precprime(n+3));if(pCharles R Greathouse IV, Feb 19 2013

a(n) = max{gpf(n), gpf(n+1), gpf(n+2), gpf(n+3)} = gpf(A052762(n+3)) with gpf = A006530, greatest prime factor.

a(n) > 47 for n > 17575. - Charles R Greathouse IV, Feb 19 2013

A230339 Numerator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Original entry on oeis.org

0, 1, 1, 19, 17, 55, 83, 119, 82, 73, 95, 121, 227, 559, 679, 815, 484, 1139, 443, 171, 295, 2023, 2299, 2599, 1462, 3275, 3653, 451, 749, 551, 5455, 5983, 3272, 7139, 7769, 8435, 1523, 3293, 3553, 11479, 6170, 13243, 14189, 15179, 8107, 5765

Offset: 0

Jean-François Alcover, Oct 16 2013

			1/(1*2*3*4) + 1/(2*3*4*5) + 1/(3*4*5*6) = 19/360, so a(3) = 19.
The rationals r(n) = a(n)/A230340(n) begin: 0, 1/24, 1/20, 19/360, 17/315, 55/1008, 83/1512, 119/2160, 82/1485, 73/1320, 95/1716, 121/2184, 227/4095, 559/10080, 679/12240, 815/14688, ... - _Wolfdieter Lang_, Mar 08 2018

L. B. W. Jolley, Summation of Series, Second revised ed., Dover, 1961, p.38, (202) and (201).

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Pochhammer Symbol

Cf. A001563, A052762, A094258, A125650, A230328, A230340 (denominators).

a[n_] := Numerator[1/18 - 1/(3*(n+1)*(n+2)*(n+3))]; Table[a[n], {n, 0, 100}]

a(n) = numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) \\ Colin Barker, Jul 30 2019

Numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) (from the generic formula Sum_{k=1..n} 1/Pochhammer(k, m) = 1/((m-1)*(m-1)!) - 1/((m-1)*Pochhammer(n+1, m-1)) with m = 4).
G.f. for the rationals r(n) = (1/18)*n*(11+n^2+6*n)/((1+n)*(n+2)*(n+3)) = a(n)/A230340(n): (1/18)*(1 - hypergeometric([1, 3], [4], -x/(1-x)))/(1-x) = (6*x - 15*x^2 + 11*x^3 +  6*(1 - 3*x + 3*x^2 - x^3)*log(1-x))/(36*x^3*(1-x)). - Wolfdieter Lang, Mar 08 2018
a(n) = numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))). - Colin Barker, Jul 30 2019

A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 2, 1, 0, 1, 10, 3, 7, 3, 0, 1, 15, 4, 25, 12, 2, 0, 1, 21, 5, 65, 30, 6, 1, 0, 1, 28, 6, 140, 60, 12, 15, 7, 0, 1, 36, 7, 266, 105, 20, 90, 50, 12, 0, 1, 45, 8, 462, 168, 30, 350, 195, 60, 6, 0, 1, 55, 9, 750, 252, 42, 1050, 560, 180, 24, 1, 0

Offset: 0

Peter Luschny, Oct 20 2017

			Array starts:
m\j| 0   1  2     3       4       5       6       7       8       9      10
---|-----------------------------------------------------------------------
m=0| 1,  0, 0,    0,      0,      0,      0,      0,      0,      0,      0
m=1| 1,  1, 1,    1,      3,      2,      1,      7,     12,      6,      1
m=2| 1,  3, 2,    7,     12,      6,     15,     50,     60,     24,     31
m=3| 1,  6, 3,   25,     30,     12,     90,    195,    180,     60,    301
m=4| 1, 10, 4,   65,     60,     20,    350,    560,    420,    120,   1701
m=5| 1, 15, 5,  140,    105,     30,   1050,   1330,    840,    210,   6951
m=6| 1, 21, 6,  266,    168,     42,   2646,   2772,   1512,    336,  22827
m=7| 1, 28, 7,  462,    252,     56,   5880,   5250,   2520,    504,  63987
m=8| 1, 36, 8,  750,    360,     72,  11880,   9240,   3960,    720, 159027
m=9| 1, 45, 9, 1155,    495,     90,  22275,  15345,   5940,    990, 359502
   A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
.
m\j| ...      11      12      13      14
---|-----------------------------------------
m=0| ...,      0,      0,      0,      0, ... [A000007]
m=1| ...,     15,     50,     60,     24, ... [A028246]
m=2| ...,    180,    390,    360,    120, ... [A053440]
m=3| ...,   1050,   1680,   1260,    360, ... [A294032]
m=4| ...,   4200,   5320,   3360,    840, ...
m=5| ...,  13230,  13860,   7560,   1680, ...
m=6| ...,  35280,  31500,  15120,   3024, ...
m=7| ...,  83160,  64680,  27720,   5040, ...
m=8| ..., 178200, 122760,  47520,   7920, ...
m=9| ..., 353925, 218790,  77220,  11880, ...
         A293476,A293608,A293615,A052762, ...
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A053440, T(3, 2) is row 2 of A294032 (which is [25, 30, 12]) and T(3, 2, 1) = 30.
.
Remark: To adapt the sequences A028246 and A053440 to our enumeration use the exponential generating functions exp(x)/(1 - y*(exp(x) - 1)) and exp(x)*(2*exp(x) - y*exp(2*x) + 2*y*exp(x) - 1 - y)/(1 - y*(exp(x) - 1))^2 instead of those indicated in their respective entries.

Eric Weisstein's World of Mathematics, Nørlund Polynomial.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
Cf. A293616.

A293617 := proc(m, n, k) option remember:
if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
# Sample uses:
A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
(* Sample use: *)
A293926Row[n_] := A293617Row[n, n];

T(m,n,k) = (k + m)*T(m, n-1, k) + k*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k<0 or k>n; and T(m, 0, k) = 0^k.

T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).

A011933 a(n) = floor( n(n-1)(n-2)*(n-3)/23 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 36, 73, 131, 219, 344, 516, 746, 1044, 1424, 1899, 2483, 3193, 4044, 5055, 6245, 7633, 9240, 11088, 13200, 15600, 18313, 21365, 24783, 28596, 32833, 37523, 42699, 48392, 54636, 61466, 68916, 77024, 85827, 95363, 105673, 116796, 128775, 141653, 155473, 170280, 186120, 203040, 221088

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Cf. A000332, A011915, A052762.

[Floor(24*Binomial(n,4)/23): n in [0..80]]; // G. C. Greubel, Nov 03 2024

Table[Floor[(n(n-1)(n-2)(n-3))/23],{n,0,60}] (* Harvey P. Dale, Jun 22 2011 *)

a(n) = n*(n-1)*(n-2)*(n-3)\23; \\ Michel Marcus, Jun 14 2017

[24*binomial(n,4)//23 for n in range(81)] # G. C. Greubel, Nov 03 2024

From Chai Wah Wu, Aug 02 2020: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-23) - 4*a(n-24) + 6*a(n-25) - 4*a(n-26) + a(n-27) for n > 26.
G.f.: x^4*(1+x^2)*(1 + x + x^3 - x^5 + 4*x^6 - x^7 - x^8 + 2*x^9 + 2*x^11 - x^12 - x^13 + 4*x^14 - x^15 + x^17 + x^19 + x^20)/((1-x)^4*(1-x^23)). (End)

More terms added by G. C. Greubel, Nov 03 2024

A158874 a(n) = (n + 4)(n + 3)(n + 2)(n + 1)n / 5 = 24*A000389(n+4).

Original entry on oeis.org

0, 24, 144, 504, 1344, 3024, 6048, 11088, 19008, 30888, 48048, 72072, 104832, 148512, 205632, 279072, 372096, 488376, 632016, 807576, 1020096, 1275120, 1578720, 1937520, 2358720, 2850120, 3420144, 4077864, 4833024, 5696064, 6678144, 7791168, 9047808

Offset: 0

N. J. A. Sloane, Nov 29 2009

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (48), page 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Partial sums of A052762.

[n*(n^4+10*n^3+35*n^2+50*n+24)/5: n in [0..30]]; // Vincenzo Librandi, Oct 05 2011

Table[(n + 4)*(n + 3)*(n + 2)*(n + 1)*n/5, {n,0,50}] (* G. C. Greubel, Nov 21 2017 *)

for(n=0,30, print1((n + 4)*(n + 3)*(n + 2)*(n + 1)*n/5, ", ")) \\ G. C. Greubel, Nov 21 2017

G.f.: 24*x / (x-1)^6 . - R. J. Mathar, Oct 03 2011
E.g.f.: x*(x^4 + 20*x^3 + 120*x^2 + 240*x + 120)*exp(x)/5. - G. C. Greubel, Nov 21 2017
From Amiram Eldar, Jul 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 5/96.
Sum_{n>=1} (-1)^(n+1)/a(n) = 10*log(2)/3 - 655/288. (End)

A230340 Denominator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Original entry on oeis.org

1, 24, 20, 360, 315, 1008, 1512, 2160, 1485, 1320, 1716, 2184, 4095, 10080, 12240, 14688, 8721, 20520, 7980, 3080, 5313, 36432, 41400, 46800, 26325, 58968, 65772, 8120, 13485, 9920, 98208, 107712, 58905, 128520, 139860, 151848, 27417, 59280

Offset: 0

Jean-François Alcover, Oct 16 2013

			1/(1*2*3*4) + 1/(2*3*4*5) + 1/(3*4*5*6) = 19/360, so a(3) = 360.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Pochhammer Symbol

Cf. A001563, A052762, A094258, A125650, A230328, A230339 (numerators).

[Denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))):n in [0..100]]; // Marius A. Burtea, Jul 30 2019

a[n_] := Denominator[1/18 - 1/(3*(n+1)*(n+2)*(n+3))]; Table[a[n], {n, 0, 100}]

a(n) = denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) \\ Colin Barker, Jul 30 2019

Denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))).

A351697 32*a(n) is the denominator of the squared circumradius of a cyclic quadrilateral with sides n, n+1, n+2, n+3.

Original entry on oeis.org

3, 15, 5, 105, 210, 42, 630, 990, 165, 2145, 3003, 455, 5460, 7140, 1020, 11628, 14535, 1995, 21945, 26565, 3542, 37950, 44850, 5850, 61425, 71253, 9135, 94395, 107880, 13640, 139128, 157080, 19635, 198135, 221445, 27417, 274170, 303810, 37310, 370230, 407253, 49665

Offset: 1

Hugo Pfoertner, Feb 26 2022

			(1/32)*{385/3, 3289/15, 1729/5, 53041/105, 146329/210, 38665/42, ...}

A351696 gives the corresponding numerators.

Cf. A000332, A052762, A345665, A351698.

a(n) = denominator(b(n)), where b(n) = (8*(2*n^6 + 18*n^5 + 65*n^4 + 120*n^3 + 117*n^2 + 54*n + 9))/(n^4 + 6*n^3 + 11*n^2 + 6*n).

A054777 a(n) = 4n(4n-1)(4n-2)(4*n-3).

Original entry on oeis.org

0, 24, 1680, 11880, 43680, 116280, 255024, 491400, 863040, 1413720, 2193360, 3258024, 4669920, 6497400, 8814960, 11703240, 15249024, 19545240, 24690960, 30791400, 37957920, 46308024, 55965360, 67059720, 79727040, 94109400, 110355024, 128618280, 149059680, 171845880

Offset: 0

Henry Bottomley, May 19 2000

L. B. W. Jolley, Summation of Series, Dover, 1961.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 268.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A052762, A060541, A104188.

[4*n*(4*n-1)*(4*n-2)*(4*n-3): n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

a[n_] := 4*n*(4*n-1)*(4*n-2)*(4*n-3); Array[a, 40, 0] (* Amiram Eldar, Mar 08 2022 *)

a(n) = A052762(4n) = 24*A060541(n).
Sum_{n>=1} 1/a(n) = log(2)/4 - Pi/24 = 0.0423871012404116... [Jolley eq. 242] - Benoit Cloitre, Apr 05 2002
G.f. -24*x*(1 + 65*x + 155*x^2 + 35*x^3) / (x-1)^5. - R. J. Mathar, Oct 03 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = log(sqrt(2)-1)/(6*sqrt(2)) - log(2)/24 + (1/(6*sqrt(2)) - 1/16)*Pi. - Amiram Eldar, Mar 08 2022

A117465 Denominator of -16/((n+2)n(n-2)*(n-4)).

Original entry on oeis.org

9, 0, 15, 0, 105, 24, 945, 120, 3465, 360, 9009, 840, 19305, 1680, 36465, 3024, 62985, 5040, 101745, 7920, 156009, 11880, 229425, 17160, 326025, 24024, 450225, 32760, 606825, 43680, 801009, 57120, 1038345, 73440, 1324785, 93024, 1666665, 116280

Offset: 1

Steven J. Forsberg, Apr 25 2006

I came up with the equation to help analyze the path to stable orbits of the logistic function
f(n+1) = k*n(1-n) for f(n) with n => 9, then f(n)*A072346(n-5) = A072346(n+3).
a(n) is the denominator of f(n). The numerator of f(n) is -1 if n is even, else -16.

			f(5) = -16/(7*5*3*1) = -16/105, denominator a(5) = 105.
f(6) = -16/(8*6*4*2) = -1/24, denominator a(6) = 24.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A052762, A072346.

f(n) := n -> (1/((n/4)+(n^2/4)-(n^3/16)-1))/n;

Join[{9,0,15,0},Denominator[Table[-(16/(n (n^3-4 n^2-4 n+16))), {n,5,40}]]]    (* Harvey P. Dale, Nov 06 2011 *)

Vec(3*x*(10*x^12-50*x^10+105*x^8-160*x^6-8*x^5-40*x^4+10*x^2-3)/((x-1)^5*(x+1)^5) + O(x^100)) \\ Colin Barker, Nov 11 2014

a(n) = denominator of the reduced -16/(n*(n-2)*(n+2)*(n-4)).
a(2n) = A052762(n+1).
a(n) = 5*a(n-2) -10*a(n-4) +10*a(n-6) -5*a(n-8) +a(n-10) for n>15. - R. J. Mathar, Mar 27 2010
a(n) = -(-17+15*(-1)^n)*(n*(16-4*n-4*n^2+n^3))/32 for n>3. - Colin Barker, Nov 11 2014
G.f.: 3*x*(10*x^12-50*x^10+105*x^8-160*x^6-8*x^5-40*x^4+10*x^2-3) / ((x-1)^5*(x+1)^5). - Colin Barker, Nov 11 2014

Clearer definition from R. J. Mathar, Mar 27 2010

cp's OEIS Frontend

A001094 a(n) = n + n*(n-1)*(n-2)*(n-3).

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A190136 Largest prime factor of n*(n+1)*(n+2)*(n+3).

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A230339 Numerator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

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A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

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A011933 a(n) = floor( n*(n-1)*(n-2)*(n-3)/23 ).

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A158874 a(n) = (n + 4)*(n + 3)*(n + 2)*(n + 1)*n / 5 = 24*A000389(n+4).

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A230340 Denominator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

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A001094 a(n) = n + n(n-1)(n-2)*(n-3).

A190136 Largest prime factor of n(n+1)(n+2)*(n+3).

A011933 a(n) = floor( n(n-1)(n-2)*(n-3)/23 ).

A158874 a(n) = (n + 4)(n + 3)(n + 2)(n + 1)n / 5 = 24*A000389(n+4).

A054777 a(n) = 4n(4n-1)(4n-2)(4*n-3).

A117465 Denominator of -16/((n+2)n(n-2)*(n-4)).