A000579 -id:A000579 - cp's OEIS frontend

A213745 Triangle of numbers C^(6)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 6 appearances allowed.

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 35, 1, 5, 15, 35, 70, 126, 1, 6, 21, 56, 126, 252, 462, 1, 7, 28, 84, 210, 462, 924, 1709, 1, 8, 36, 120, 330, 792, 1716, 3424, 6371, 1, 9, 45, 165, 495, 1287, 3003, 6426, 12789, 23905, 1, 10

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012

For k<=5, the triangle coincides with triangle A213744.

We have over columns of the triangle: T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n>1, T(n,3)=A000292(n) for n>=3, T(n,4)=A000332(n) for n>=7, T(n,5)=A000389(n) for n>=9, T(n,6)=A000579(n) for n>=11, T(n,7)=A063267 for n>=5, T(n,8)=A063417 for n>=6, T(n,9)=A063418 for n>=7.

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....3
.3..|..1.....3.....6....10
.4..|..1.....4....10....20....35
.5..|..1.....5....15....35....70....126
.6..|..1.....6....21....56...126....252...462
.7..|..1.....7....28....84...210....462...924....1709

Peter J. C. Moses, Rows n = 0..50 of triangle, flattened

Cf. A007318, A005725, A059481, A111808, A187925, A213742, A213743, A213744, A000217, A000292, A000332, A000389, A000579, A063267, A063417, A063418.

Flatten[Table[Sum[(-1)^r Binomial[n,r] Binomial[n-# r+k-1,n-1],{r,0,Floor[k/#]}],{n,0,15},{k,0,n}]/.{0}->{1}]&[7] (* Peter J. C. Moses, Apr 16 2013 *)

C^(6)(n,k)=sum{r=0,...,floor(k/7)}(-1)^r*C(n,r)*C(n-7*r+k-1, n-1).
A generalization. The numbers C^(t)(n,k) of combinations with repetitions from n different elements over k, for each of them not more than t>=1 appearances allowed, are enumerated by the formula:
C^(t)(n,k)=sum{r=0,...,floor(k/(t+1))}(-1)^r*C(n,r)*C(n-(t+1)*r+k-1, n-1).
In case t=1, it is binomial coefficient C^(t)(n,k)=C(n,k), and we have the combinatorial identity: sum{r=0,...,floor(k/2)}(-1)^r*C(n,r)*C(n-2*r+k-1, n-1)=C(n,k). On the other hand, if t=n, then r=0, and for the corresponding numbers of combinations with repetitions without a restriction on appearances of elements we obtain a well known formula C(n+k-1, n-1) (cf. triangle A059481).
In addition, note that, if k<=t, then C^(t)(n,k)=C(n+k-1, n-1). Therefore, triangle {C^(t+1)(n,k)} coincides with the previous triangle {C^(t)(n,k)} for k<=t.

A299338 Expansion of 1 / ((1 - x)^7*(1 + x)^6).

Original entry on oeis.org

1, 1, 7, 7, 28, 28, 84, 84, 210, 210, 462, 462, 924, 924, 1716, 1716, 3003, 3003, 5005, 5005, 8008, 8008, 12376, 12376, 18564, 18564, 27132, 27132, 38760, 38760, 54264, 54264, 74613, 74613, 100947, 100947, 134596, 134596, 177100, 177100, 230230, 230230

Offset: 0

Colin Barker, Feb 07 2018

Same as A000579 but with repeated terms.

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

Cf. A000579, A001769, A060099, A299335, A299336, A299337.

CoefficientList[Series[1/((1-x)^7(1+x)^6),{x,0,50}],x] (* or *) LinearRecurrence[ {1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,1,7,7,28,28,84,84,210,210,462,462,924},50] (* Harvey P. Dale, Oct 09 2018 *)

Vec(1 / ((1 - x)^7*(1 + x)^6) + O(x^40))

a(n) = (2*n^6 + 84*n^5 + 1400*n^4 + 11760*n^3 + 51968*n^2 + 112896*n + 92160) / 92160 for n even.
a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 92160 for n odd.
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>12.

A344206 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+5,6).

Original entry on oeis.org

1, 7, 28, 112, 210, 658, 924, 2388, 3409, 6475, 8008, 18746, 18564, 33600, 44640, 72408, 74613, 137746, 134596, 235655, 256102, 352066, 376740, 680260, 615930, 866229, 994336, 1400980, 1344904, 2172800, 1947792, 2929332, 2984905, 3784914, 4032420, 6128858, 5245786

Offset: 1

Ilya Gutkovskiy, May 11 2021

nonn

Cf. A000428, A000579, A001055, A050367, A329124, A344203, A344204, A344205, A344207.

A063267 Eighth column (k=7) of septinomial array A063265.

Original entry on oeis.org

6, 33, 116, 325, 786, 1709, 3424, 6426, 11430, 19437, 31812, 50375, 77506, 116265, 170528, 245140, 346086, 480681, 657780, 888009, 1184018, 1560757, 2035776, 2629550, 3365830, 4272021, 5379588, 6724491, 8347650

Offset: 0

Wolfdieter Lang, Jul 24 2001

Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A000579 (column k=6 of A063265).

[seq((binomial(n+7,n)-binomial(n+1,n)),n=1..29)]; # Zerinvary Lajos, Jun 23 2006

Table[Binomial[n+7,n]-Binomial[n+1,n],{n,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{6,33,116,325,786,1709,3424,6426},30] (* Harvey P. Dale, Jan 06 2012 *)

a(n)= A063265(n+2, 7)= (n+1)*(n+2)*(n+10)*(n^4 + 22*n^3 + 193*n^2 + 792*n + 1512)/7!.
G.f.: (2-x)*(1-x+x^2)*(3-3*x+x^2)/(1-x)^8; the numerator polynomial is N7(7, x) = 6 - 15*x + 20*x^2 - 15*x^3 + 6*x^4 - x^5 from row n=7 of array A063266.
a(n) = binomial(n+7,n) - binomial(n+1,n). - Zerinvary Lajos, Jun 23 2006
a(n) = binomial(n+7,n) + binomial(n+6,n) + binomial(n+5,n) + binomial(n+4,n) + binomial(n+3,n) + binomial(n+2,n). - Zerinvary Lajos, Jun 23 2006
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8); a(0)=6, a(1)=33, a(2)=116, a(3)=325, a(4)=786, a(5)=1709, a(6)=3424, a(7)=6426. - Harvey P. Dale, Jan 06 2012

More terms from Zerinvary Lajos, Jun 23 2006

A081905 Third binomial transform of binomial(n+6, 6).

Original entry on oeis.org

1, 10, 79, 552, 3567, 21810, 127905, 725820, 4009920, 21664000, 114840064, 598865920, 3078537216, 15626600448, 78431059968, 389685706752, 1918516592640, 9367021682688, 45387134009344, 218388081147904, 1044061452500992, 4961718019031040, 23449374679891968, 110252343064264704

Offset: 0

Paul Barry, Mar 31 2003

Binomial transform of A081904.

4th binomial transform of (1,6,15,20,15,6,1,0,0,0,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (28,-336,2240,-8960,21504,-28672,16384).

Cf. A000579, A081904, A081906.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)^6/(1-4*x)^7)); // G. C. Greubel, Oct 17 2018

CoefficientList[Series[(1-3x)^6/(1-4x)^7,{x,0,20}],x] (* or *) LinearRecurrence[{28,-336,2240,-8960,21504,-28672,16384},{1,10,79,552,3567,21810,127905},20] (* Harvey P. Dale, Aug 14 2014 *)

x='x+O(x^30); Vec((1-3*x)^6/(1-4*x)^7) \\ G. C. Greubel, Oct 17 2018

a(n) = 4^n*(n^6 + 129*n^5 + 5845*n^4 + 115215*n^3 + 993874*n^2 + 3308616*n + 2949120)/2949120.
G.f.: (1-3*x)^6/(1-4*x)^7.
a(n) = 28*a(n-1) - 336*a(n-2) + 2240*a(n-3) - 8960*a(n-4) + 21504*a(n-5) - 28672*a(n-6) + 16384*a(n-7); a(0)=1, a(1)=10, a(2)=79, a(3)=552, a(4)=3567, a(5)=21810, a(6)=127905. - Harvey P. Dale, Aug 14 2014
E.g.f.: (720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(4*x) / 720. - G. C. Greubel, Oct 17 2018

A095662 Seventh column (m=6) of (1,3)-Pascal triangle A095660.

Original entry on oeis.org

3, 19, 70, 196, 462, 966, 1848, 3300, 5577, 9009, 14014, 21112, 30940, 44268, 62016, 85272, 115311, 153615, 201894, 262108, 336490, 427570, 538200, 671580, 831285, 1021293, 1246014, 1510320, 1819576, 2179672, 2597056, 3078768, 3632475

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 3-subset of an n-set X then, for n>=8, a(n-8) is the number of 6-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Sixth column: A000574. Eighth column: A095663.

CoefficientList[Series[(3-2x)/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{3,19,70,196,462,966,1848},40] (* Harvey P. Dale, Mar 30 2014 *)

G.f.: (3-2*x)/(1-x)^7.
a(n)= binomial(n+5, 5)*(n+18)/6 = 3*b(n)-2*b(n-1), with b(n):=binomial(n+6, 6); cf. A000579.
a(0)=3, a(1)=19, a(2)=70, a(3)=196, a(4)=462, a(5)=966, a(6)=1848, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Mar 30 2014

A105939 a(n) = binomial(n+3,3)*binomial(n+6,3).

Original entry on oeis.org

20, 140, 560, 1680, 4200, 9240, 18480, 34320, 60060, 100100, 160160, 247520, 371280, 542640, 775200, 1085280, 1492260, 2018940, 2691920, 3542000, 4604600, 5920200, 7534800, 9500400, 11875500, 14725620, 18123840, 22151360, 26898080, 32463200, 38955840, 46495680

Offset: 0

Zerinvary Lajos, Apr 27 2005

a(n) is the number of ordered pairs (A,B) of size 3 subsets of {1,2,...,n+6} such that A and B are disjoint. - Geoffrey Critzer, Sep 03 2013

			If n=0 then C(0+3,0)*C(0+6,3) = C(3,0)*C(6,3) = 1*20 = 20.
If n=8 then C(8+3,8)*C(8+6,3) = C(11,8)*C(14,3) = 165*364 = 60060.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000292, A062145.

A105939:= func< n | 20*Binomial(n+6,6) >;
[A105939(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

nn=25; f[x_]:=Exp[x](x^3/3!)^2;Range[0,nn]! CoefficientList[Series[ a=f''''''[x],{x,0,nn}],x] (* Geoffrey Critzer, Sep 03 2013 *)
Table[Binomial[n+3,3]Binomial[n+6,3],{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{20,140,560,1680,4200,9240,18480},30] (* Harvey P. Dale, Mar 09 2022 *)
20*Binomial[Range[0,40] +6,6] (* G. C. Greubel, Mar 11 2025 *)

def A105939(n): return 20*binomial(n+6,6)
print([A105939(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

G.f.: 20/(1-x)^7. - Colin Barker, Jun 06 2012
E.g.f.: (d/dx)^6 (x^3/3!)^2 * exp(x). - Geoffrey Critzer, Sep 03 2013
a(n) = A000292(n+1)*A000292(n+4) = 20*A000579(n+6). - R. J. Mathar, Nov 30 2015
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 3/50.
Sum_{n>=0} (-1)^n/a(n) = 48*log(2)/5 - 661/100. (End)
E.g.f.: (1/36)*(720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(x). - G. C. Greubel, Mar 11 2025

More terms from Geoffrey Critzer, Sep 03 2013

A115567 a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 126, 246, 465, 847, 1485, 2509, 4095, 6475, 9948, 14892, 21777, 31179, 43795, 60459, 82159, 110055, 145498, 190050, 245505, 313911, 397593, 499177, 621615, 768211, 942648, 1149016, 1391841, 1676115, 2007327, 2391495

Offset: 0

Jonathan Vos Post, Mar 12 2006

a(n) = n + T(n) + Tet(n) + Ptop(n) + 5-Simplex(n) + 6-Simplex(n), where T(n) = n-th triangular number A000217(n), Tet(n) = n-th tetrahedral number A000292(n), Ptop(n) = n-th pentatope number A000332(n), 5-Simplex(n) = n-th 5-simplex number A000389(n), 6-Simplex(n) = n-th 6-simplex number A000579(n).

By analogy to A004006, A055795 and A057703, I presume that a(n) = Answer to the question: if you have a tall building and 6 plates and you need to find the highest story, a plate thrown from which does not break, what is the number of stories you can handle given n tries?

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [_Parthasarathy Nambi_, Sep 30 2009]
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000217, A000332, A000389, A000292, A000579, A004006, A055795, A057703.

[n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720: n in [0..30]]; // G. C. Greubel, Nov 25 2017

seq(sum(binomial(n,k),k=1..6),n=0..36); # Zerinvary Lajos, Dec 13 2007

Table[n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720, {n,0,30}] (* G. C. Greubel, Nov 25 2017 *)

for(n=0,30, print1(n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720, ", ")) \\ G. C. Greubel, Nov 25 2017

[binomial(n,2)+binomial(n,4)+binomial(n,6) for n in range(1, 38)] # Zerinvary Lajos, May 17 2009

[binomial(n,1)+binomial(n,3)+binomial(n,5)+binomial(n,2)+binomial(n,4)+binomial(n,6) for n in range(0, 37)] # Zerinvary Lajos, May 17 2009

a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).
a(n) = A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.
a(n) = A000579(n) + A057703(n).
G.f.: x*(1-x+x^2)*(1-3*x+3*x^2)/(1-x)^7. - Colin Barker, Mar 16 2012
From G. C. Greubel, Nov 25 2017: (Start)
a(n) = n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720.
E.g.f.: x*(720 + 360*x + 120*x^2 + 30*x^3 + 6*x^4 + x^5)*exp(x)/720. (End)

A124089 a(n) = binomial(n,6)-1.

Original entry on oeis.org

0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, 18563, 27131, 38759, 54263, 74612, 100946, 134595, 177099, 230229, 296009, 376739, 475019, 593774, 736280, 906191, 1107567, 1344903, 1623159, 1947791, 2324783, 2760680, 3262622

Offset: 6

Zerinvary Lajos, Nov 25 2006

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A000096, A000579, A062748, A063258, A062988.

[Binomial(n,6)-1 : n in [6..40]]; // Wesley Ivan Hurt, Dec 27 2023

Maple
```
[seq(binomial(n,6)-1,n=6..42)];
```

Binomial[Range[6,40],6]-1 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,6,27,83,209,461,923},40] (* Harvey P. Dale, Dec 26 2015 *)

a(n) = A000579(n)-1.

a(0)=0, a(1)=6, a(2)=27, a(3)=83, a(4)=209, a(5)=461, a(6)=923, a(n)= 7*a(n-1)- 21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+ a(n-7). - Harvey P. Dale, Dec 26 2015

A247976 Triangle read by rows: T(n,k) generated by m-gon expansions in the case of odd m with "vertex to vertex" version or even m with "vertex to side" version. (See comment for details.)

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7

Offset: 1

Kival Ngaokrajang, Sep 28 2014

Refer to triangle expansions in A061777 and A101946 (and their companions for m-gons) which are "vertex to vertex" and "vertex to side" versions respectively. The label values at each iteration can be arranged as triangle. Any m-gon can also be arranged as the same triangle with conditions: (i) m is odd and expansion is "vertex to vertex" version or (ii) m is even and expansion is "vertex to side" version. m*Sum_{i=1..k}T(n,k) gives the total label value in n-th iteration. See illustration.

			Triangle begins:
  1;
  1,  1;
  1,  1,  2;
  1,  2,  1,  2;
  1,  2,  1,  3,  3;
  1,  3,  3,  1,  3,  3;
  1,  3,  3,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  5, 10, 10,  5;
  1,  5, 10, 10,  5,  1,  5, 10, 10, 5;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Kival Ngaokrajang, Illustration of initial terms

Rows sum: A027383.
Column (start from 1s): c3=A008805, c4=A058187, c5=A000332 repeated, c6=A000389 repeated, c7=A000579 repeated.
Vertex to vertex: A061777, A247618, A247619, A247620.
Vertex to side: A101946, A247903, A247904, A247905.
Cf. A074909.

T[n_, k_]:= T[n, k]= If[k==1, 1, If[k==n, Floor[(n+1)/2], If[OddQ[n], If[k<=(n+ 1)/2, T[n-1, k], T[n-1, k-1] + T[n-1, k]], If[kG. C. Greubel, Feb 18 2022 *)

@CachedFunction
def T(n,k): # A247976
    if (k==1): return 1
    elif (k==n): return (n+1)//2
    elif (n%2==1): return T(n-1,k) if (k <= (n+1)/2) else T(n-1,k-1) + T(n-1,k)
    else: return T(n-1,k-1)+T(n-1,k) if (k < (n+2)/2) else T(n,k-n/2)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Feb 18 2022

T(n, k) = ( T(n-1, k) if k <= (n+1)/2 otherwise T(n-1, k-1) + T(n-1, k) ) for odd n rows, ( T(n-1, k-1) + T(n-1, k) if k < (n+2)/2 otherwise T(n, k - n/2) ) for even n rows, with T(n, 1) = 1 and T(n, n) = floor((n+1)/2). - G. C. Greubel, Feb 18 2022

cp's OEIS Frontend

A213745 Triangle of numbers C^(6)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 6 appearances allowed.

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A299338 Expansion of 1 / ((1 - x)^7*(1 + x)^6).

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A344206 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+5,6).

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A063267 Eighth column (k=7) of septinomial array A063265.

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Maple

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A081905 Third binomial transform of binomial(n+6, 6).

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A095662 Seventh column (m=6) of (1,3)-Pascal triangle A095660.

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A105939 a(n) = binomial(n+3,3)*binomial(n+6,3).

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A115567 a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

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