A000389 -id:A000389 - cp's OEIS frontend

A306462 Number of ways to write n as C(2w,2) + C(x+2,3) + C(y+3,4) + C(z+4,5), where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!), w is a positive integer and x,y,z are nonnegative integers.

1, 3, 3, 1, 1, 4, 7, 6, 2, 2, 6, 8, 5, 1, 2, 9, 11, 5, 1, 4, 9, 12, 7, 2, 4, 10, 12, 7, 4, 6, 10, 11, 6, 5, 5, 10, 15, 8, 4, 7, 11, 14, 9, 4, 5, 11, 14, 6, 6, 10, 15, 12, 5, 7, 8, 11, 14, 7, 5, 6, 11, 14, 12, 11, 6, 11, 15, 12, 7, 9, 18, 21, 12, 5, 5, 15, 19, 11, 3, 5

Offset: 1

Zhi-Wei Sun, Feb 17 2019

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 4, 5, 14, 19.
We have verified a(n) > 0 for all n = 1..5*10^6.
See also A306471 and A306477 for similar conjectures.

			a(1) = 1 with 1 = C(2,2) + C(2,3) + C(3,4) + C(4,5).
a(4) = 1 with 4 = C(2,2) + C(3,3) + C(4,4) + C(5,5).
a(5) = 1 with 5 = C(2,2) + C(4,3) + C(3,4) + C(4,5).
a(14) = 1 with 14 = C(4,2) + C(3,3) + C(4,4) + C(6,5).
a(19) = 1 with 19 = C(6,2) + C(4,3) + C(3,4) + C(4,5).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000217, A000292, A000332, A000384, A000389, A000797, A262813, A306460, A306471, A306477.

f[m_,n_]:=f[m,n]=Binomial[m+n-1,m];
HQ[n_]:=HQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==3;
tab={};Do[r=0;Do[If[f[5,z]>=n,Goto[cc]];Do[If[f[4,y]>=n-f[5,z],Goto[bb]];Do[If[f[3,x]>=n-f[5,z]-f[4,y],Goto[aa]];If[HQ[n-f[5,z]-f[4,y]-f[3,x]],r=r+1],{x,0,n-1-f[5,z]-f[4,y]}];Label[aa],{y,0,n-1-f[5,z]}];Label[bb],{z,0,n-1}];Label[cc];tab=Append[tab,r],{n,1,80}];Print[tab]

A306471 Number of ways to write n as C(2w+1,2) + C(x+2,3) + C(y+3,4) + C(z+4,5) with w,x,y,z nonnegative integers, where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!).

Original entry on oeis.org

1, 3, 3, 2, 4, 6, 5, 4, 4, 5, 7, 8, 6, 4, 5, 8, 8, 5, 4, 6, 7, 10, 10, 6, 6, 12, 13, 8, 7, 7, 6, 11, 9, 4, 3, 8, 16, 12, 8, 9, 9, 13, 14, 10, 7, 9, 18, 12, 6, 5, 4, 11, 10, 4, 2, 5, 19, 21, 11, 9, 13, 20, 16, 9, 6, 8, 17, 17, 4, 2, 9, 20, 17, 6, 9, 9, 15, 23, 14, 9, 15

Offset: 0

Zhi-Wei Sun, Feb 17 2019

nonn

Conjecture 1: a(n) > 1 for all n > 0.
We have verified a(n) > 0 for all n = 0..5*10^6.
Conjecture 2: For each r = 0, 1, any positive integer can be written as w^2 + C(x,3) + C(y,4) + C(z,5), where w,x,y,z are nonnegative integers with w - r even.
See also A306462 and A306477 for similar conjectures.

			a(0) = 1 with 0 = C(1,2) + C(2,3) + C(3,4) + C(4,5).
a(3) = 2 with 3 = C(3,2) + C(2,3) + C(3,4) + C(4,5) = C(1,2) + C(3,3) + C(4,4) + C(5,5).
a(54) = 2 with 54 = C(3,2) + C(7,3) + C(6,4) + C(5,5) = C(3,2) + C(5,3) + C(7,4) + C(6,5).
a(69) = 1 with 69 = C(3,2) + C(5,3) + C(7,4) + C(7,5) = C(3,2) + C(5,3) + C(3,4) + C(8,5).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A000292, A000332, A000389, A000797, A014105, A262813, A306460, A306462, A306477.

f[m_,n_]:=f[m,n]=Binomial[m+n-1,m];
HQ[n_]:=HQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==1;
tab={};Do[r=0;Do[If[f[5,z]>n,Goto[cc]];Do[If[f[4,y]>n-f[5,z],Goto[bb]];Do[If[f[3,x]>n-f[5,z]-f[4,y],Goto[aa]];If[HQ[n-f[5,z]-f[4,y]-f[3,x]],r=r+1],{x,0,n-f[5,z]-f[4,y]}];Label[aa],{y,0,n-f[5,z]}];Label[bb],{z,0,n}];Label[cc];tab=Append[tab,r],{n,0,80}];Print[tab]

A323533 a(n) = Product_{k=1..n} (binomial(k-1,5) + binomial(n-k,5)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 790420571136, 100389735898841088, 14582663231533605863424, 2458550581659926554038239232, 529554691027323329170207744475136, 146980847512952623091566575072055001088, 53003014923687519392206631372837133989462016

Offset: 0

Vaclav Kotesovec, Jan 17 2019

nonn

Cf. A000389, A323425, A323496, A323497, A323534, A323535.

Table[Product[Binomial[k-1,5] + Binomial[n-k,5], {k, 1, n}], {n, 0, 20}]

a(n) = prod(k=1, n, binomial(k-1,5) + binomial(n-k,5)); \\ Michel Marcus, Jan 17 2019

a(n) ~ exp((1 + 2*Pi*sqrt(5 - 2/sqrt(5))/5) * (n-5)) * n^(5*n) / (exp(5*n)*120^n).

A338981 Number of unoriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 1, 92307499707443390526727850063502, 124792381938502167392061689732085833655832902312754962, 122697712831831745940423467267565845711242845618544066030140191642464

Offset: 0

Robert A. Russell, Dec 13 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>120, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is bp(20)/15 + bp(30)/10 + bp(40)/15 + bp(50)/12 + 43*bp(60)/300 + bp(66)/10 + bp(100)/360 + bp(104)/9 + bp(114)/12 + 13*bp(120)/300 + bp(150)/240 + bp(152)/8 + bp(200)/360 + bp(208)/36 + 61*bp(300)/14400 + bp(302)/32 + bp(330)/240 + bp(600)/14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(24)/15 + bp(36)/10 + bp(48)/15 + bp(60)/12 + 7*bp(72)/300 + 3*bp(76)/25 + bp(84)/10 + 41*bp(120)/360 + bp(132)/12 + 7*bp(144)/300 + bp(152)/50 + bp(180)/240 + bp(182)/8 + 11*bp(240)/360 + 61*bp(360)/14400 + bp(364)/32 + bp(396)/240 + bp(720)/14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(40)/15 + bp(60)/10 + bp(80)/15 + bp(100)/12 + 43*bp(120)/300 + bp(128)/10 + bp(200)/360 + bp(202)/9 + bp(216)/12 + 13*bp(240)/300 + bp(300)/240 + bp(302)/8 + bp(400)/360 + bp(404)/36 + 61*bp(600)/14400 + bp(604)/32 + bp(640)/240 + bp(1200)/14400.

Robert A. Russell, Table of n, a(n) for n = 0..120

Cf. A338980 (oriented), A338982 (chiral), A338983 (achiral), A338965 (up to n colors), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (24-cell).

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[bp[4]/15+bp[6]/10+bp[8]/15+bp[10]/12+7bp[12]/300+bp[16]/50+bp[17]/10+bp[19]/10+bp[20]/360+bp[22]/36+bp[23]/12+7bp[24]/300+bp[27]/12+bp[30]/240+bp[31]/8+bp[32]/50+bp[40]/360+bp[44]/36+bp[60]/14400+bp[61]/240+bp[62]/32+bp[75]/240+bp[120]/14400,x]

A338965(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338980(n) - A338982(n) = (A338980(n) + A338983(n)) / 2 = A338982(n) + A338983(n).
G.f.: bp(4)/15 + bp(6)/10 + bp(8)/15 + bp(10)/12 + 7bp(12)/300 + bp(16)/50 + bp(17)/10 + bp(19)/10 + bp(20)/360 + bp(22)/36 + bp(23)/12 + 7bp(24)/300 + bp(27)/12 + bp(30)/240 + bp(31)/8 + bp(32)/50 + bp(40)/360 + bp(44)/36 + bp(60)/14400 + bp(61)/240 + bp(62)/32 + bp(75)/240 + bp(120)/14400, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

A338982 Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 0, 92307499707128546879177569498768, 124792381938502167386992798774696507063550726794469211, 122697712831831745940423455373835049129541140194826165569091574960692

Offset: 0

Robert A. Russell, Dec 13 2020

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>120, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is bp(20)/15 + bp(30)/10 + bp(40)/15 + bp(50)/12 - 17*bp(60)/300 - bp(66)/10 + bp(100)/360 - bp(104)/18 - bp(114)/12 + 13*bp(120)/300 + bp(150)/240 - bp(152)/8 + bp(200)/360 + bp(208)/36 - 59*bp(300)/14400 + bp(302)/32 - bp(330)/240 + bp(600)/14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(24)/15 + bp(36)/10 + bp(48)/15 + bp(60)/12 + 7*bp(72)/300 - 2*bp(76)/25 - bp(84)/10 - 19*bp(120)/360 - bp(132)/12 + 7*bp(144)/300 + bp(152)/50 + bp(180)/240 - bp(182)/8 + 11*bp(240)/360 - 59*bp(360)/14400 + bp(364)/32 - bp(396)/240 + bp(720)/14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(40)/15 + bp(60)/10 + bp(80)/15 + bp(100)/12 - 17*bp(120)/300 - bp(128)/10 + bp(200)/360 - bp(202)/18 - bp(216)/12 + 13*bp(240)/300 + bp(300)/240 - bp(302)/8 + bp(400)/360 + bp(404)/36 - 59*bp(600)/14400 + bp(604)/32 - bp(640)/240 + bp(1200)/14400.

Robert A. Russell, Table of n, a(n) for n = 0..120

Cf. A338980 (oriented), A338981 (unoriented), A338983 (achiral), A338966 (up to n colors), A000389 (5-cell), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell).

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[bp[4]/15+bp[6]/10+bp[8]/15+bp[10]/12+7bp[12]/300+bp[16]/50-bp[17]/10-bp[19]/10+bp[20]/360+bp[22]/36-bp[23]/12+7bp[24]/300-bp[27]/12+bp[30]/240-bp[31]/8+bp[32]/50+bp[40]/360+bp[44]/36+bp[60]/14400-bp[61]/240+bp[62]/32-bp[75]/240+bp[120]/14400,x]

A338966(n) = Sum_{j=2..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338980(n) - A338981(n) = (A338980(n) - A338983(n)) / 2 = A338981(n) - A338983(n).
G.f.: bp(4)/15 + bp(6)/10 + bp(8)/15 + bp(10)/12 + 7*bp(12)/300 + bp(16)/50 - bp(17)/10 - bp(19)/10 + bp(20)/360 + bp(22)/36 - bp(23)/12 + 7*bp(24)/300 - bp(27)/12 + bp(30)/240 - bp(31)/8 + bp(32)/50 + bp(40)/360 + bp(44)/36 + bp(60)/14400 - bp(61)/240 + bp(62)/32 - bp(75)/240 + bp(120)/14400, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

A369794 Expansion of 1/(1 - x^5/(1-x)^6).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 253, 474, 870, 1651, 3367, 7372, 16762, 38183, 85290, 185573, 394555, 826752, 1724816, 3613968, 7642004, 16313856, 35052905, 75487110, 162349105, 348018300, 743376838, 1583718457, 3370144462, 7173308802, 15285181447

Offset: 0

Seiichi Manyama, Feb 01 2024

Number of compositions of 6*n-5 into parts 5 and 6.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A095263, A290998, A368475.

Cf. A000389, A099242, A017837, A107025.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^5/(1-x)^6))

a(n) = A107025(n)-A107025(n-1). First differences of A107025.
a(n) = A017837(6*n-5) = Sum_{k=0..floor((6*n-5)/5)} binomial(k,6*n-5-5*k) for n > 0.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6) for n > 6.
a(n) = Sum_{k=0..floor(n/5)} binomial(n-1+k,n-5*k).

A133084 A007318 * A133080.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 4, 1, 5, 4, 10, 4, 1, 6, 5, 20, 10, 6, 1, 7, 6, 35, 20, 21, 6, 1, 8, 7, 56, 35, 56, 21, 8, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 10, 9, 120, 84, 252, 126, 120, 36, 10, 1, 11, 10, 165, 120, 462, 252, 330, 120, 55, 10, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...).

A133084 is jointly generated with A133567 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012

			First few rows of the triangle:
  1;
  2,  1;
  3,  2,  1;
  4,  3,  4,  1;
  5,  4, 10,  4,  1;
  6,  5, 20, 10,  6,  1;
  7,  6, 35, 20, 21,  6,  1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A133080, A003945, A133567.

Cf. A000292 (column 3 and 4), A000389 (column 5 and 6), A000580 (column 7).

/* As triangle */ [[(1-(1+(-1)^k)/2 )*Binomial(n, k)+((1+(-1)^k)/2)*Binomial(n-1, k-1): k in [1..n]]: n in [1.. 11]]; // Vincenzo Librandi, Oct 21 2017

A133084 := proc(n,k)
    add(binomial(n-1,i-1)*A133080(i,k),i=1..n) ;
end proc: # R. J. Mathar, Jun 13 2025

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A133567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A133084 *)
(* Clark Kimberling, Feb 28 2012 *)
T[n_, k_] := If[k == n, 1, (1  - (1 + (-1)^k)/2 )*Binomial[n, k] + ((1 + (-1)^k)/2)*Binomial[n - 1, k - 1]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

for(n=1,10, for(k=1,n, print1(if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)), ", "))) \\ G. C. Greubel, Oct 21 2017

Binomial transform of triangle A133080.

A140404 a(n) = binomial(n+5, 5)*7^n.

Original entry on oeis.org

1, 42, 1029, 19208, 302526, 4235364, 54353838, 652246056, 7419298887, 80787921214, 848273172747, 8636963213424, 85649885199788, 830145041167176, 7886377891088172, 73606193650156272, 676256904160810749, 6126091955339109138, 54794489156088698401, 484498640959100070072

Offset: 0

Zerinvary Lajos, Jun 16 2008

With a different offset, number of n-permutations of 8 objects:r,s,t,u,v,z,x,y with repetition allowed, containing exactly five (5) u's. Example: a(1)=42 because we have
uuuuur, uuuuru, uuuruu, uuruuu, uruuuu, ruuuuu
uuuuus, uuuusu, uuusuu, uusuuu, usuuuu, suuuuu,
uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu,
uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu,
uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu,
uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu,
uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (42,-735,6860,-36015,100842,-117649).

Cf. A000389, A036084, A036219, A045543, A050982, A054849.

[7^n* Binomial(n+5, 5): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Maple
```
seq(binomial(n+5,5)*7^n,n=0..17);
```

Table[Binomial[n+5,5]7^n,{n,0,20}] (* or *) LinearRecurrence[ {42,-735,6860,-36015,100842,-117649},{1,42,1029,19208,302526,4235364},21] (* Harvey P. Dale, Sep 08 2011 *)

a(n)=binomial(n+5,5)*7^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 1/(1-7*x)^6. - Zerinvary Lajos, Aug 06 2008
a(n) = 42*a(n-1) - 735*a(n-2) + 6860*a(n-3) - 36015*a(n-4) + 100842*a(n-5) - 117649*a(n-6). - Harvey P. Dale, Sep 08 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 45360*log(7/6) - 27965/4.
Sum_{n>=0} (-1)^n/a(n) = 143360*log(8/7) - 229705/12. (End)

A185876 Fourth accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 5, 6, 15, 29, 21, 35, 85, 99, 56, 70, 195, 285, 259, 126, 126, 385, 645, 735, 574, 252, 210, 686, 1260, 1645, 1610, 1134, 462, 330, 1134, 2226, 3185, 3570, 3150, 2058, 792, 495, 1770, 3654, 5586, 6860, 6930, 5670, 3498, 1287, 715, 2640, 5670, 9114, 11956, 13230, 12390, 9570, 5643, 2002, 1001, 3795, 8415, 14070

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,   5,  15,   35,   70
   6,  29,  85,  195,  385
  21,  99, 285,  645, 1260
  56, 259, 735, 1645, 3185

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A051340, A141419, A144112, A185874, A185875.

Row 1: A000332, column 1: A000389.

f[n_,k_]:=k(1+k)n(1+n)(2+n)(5+4k+3n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185875 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185876 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185876 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = (4*n+5*k+11)*C(k+2,3)*C(n+4,4)/20, k>=1, n>=1.

A293615 a(n) = Pochhammer(n, 5) / 2.

Original entry on oeis.org

0, 60, 360, 1260, 3360, 7560, 15120, 27720, 47520, 77220, 120120, 180180, 262080, 371280, 514080, 697680, 930240, 1220940, 1580040, 2018940, 2550240, 3187800, 3946800, 4843800, 5896800, 7125300, 8550360, 10194660, 12082560, 14240160, 16695360, 19477920, 22619520

Offset: 0

Peter Luschny, Oct 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A265609, A293475, A293476, A293608.

[0] cat [Factorial(n+4)/(2*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017

A293615 := n -> pochhammer(n, 5)/2:
seq(A293615(n), n=0..11);

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 60, 360, 1260, 3360, 7560}, 32]
Table[Pochhammer[n, 5]/2, {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)

for(n=0,30, print1(n*(n+1)*(n+2)*stirling(4 + n, 3 + n, 2), ", ")) \\ G. C. Greubel, Nov 20 2017

concat(0, Vec(60*x / (1 - x)^6 + O(x^40))) \\ Colin Barker, Nov 21 2017

a(n) = n*(n+1)*(n+2)*Stirling2(4 + n, 3 + n).
-a(-n-4) = a(n) for n >= 0.
a(n) = 60*A000389(n+4). - G. C. Greubel, Nov 20 2017
From Colin Barker, Nov 21 2017: (Start)
G.f.: 60*x / (1 - x)^6.
a(n) = (1/2)*(n*(1 + n)*(2 + n)*(3 + n)*(4 + n)).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 1/48.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 131/144. (End)

cp's OEIS Frontend

A306462 Number of ways to write n as C(2w,2) + C(x+2,3) + C(y+3,4) + C(z+4,5), where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!), w is a positive integer and x,y,z are nonnegative integers.

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A306471 Number of ways to write n as C(2w+1,2) + C(x+2,3) + C(y+3,4) + C(z+4,5) with w,x,y,z nonnegative integers, where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!).

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A323533 a(n) = Product_{k=1..n} (binomial(k-1,5) + binomial(n-k,5)).

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A338981 Number of unoriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

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A338982 Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

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A369794 Expansion of 1/(1 - x^5/(1-x)^6).

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A133084 A007318 * A133080.

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A140404 a(n) = binomial(n+5, 5)*7^n.

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