A062989 -id:A062989 - cp's OEIS frontend

A063260 Sextinomial (also called hexanomial) coefficient array.

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780

Offset: 0

Wolfdieter Lang, Jul 24 2001

The sequence of step width of this staircase array is [1,5,5,...], hence the degree sequence for the row polynomials is [0,5,10,15,...]=A008587.
The column sequences (without leading zeros) are for k=0..5 those of the lower triangular array A007318 (Pascal) and for k=6..9: A062989, A063262-4. Row sums give A000400 (powers of 6). Central coefficients give A063419; see also A018901.
This can be used to calculate the number of occurrences of a given roll of n six-sided dice, where k is the index: k=0 being the lowest possible roll (i.e., n) and n*6 being the highest roll.

			The irregular table T(n, k) begins:
n\k 0 1 2  3  4  5  6  7  8  9 10 11 12 13 14 15
1:  1
2:  1 1 1  1  1  1
3:  1 2 3  4  5  6  5  4  3  2  1
4:  1 3 6 10 15 21 25 27 27 25 21 15 10  6  3  1
...reformatted - _Wolfdieter Lang_, Oct 31 2015

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.

T. D. Noe, Rows n = 0..25, flattened
Steven R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
Iain G. Johnston, Optimal strategies in the Fighting Fantasy gaming system: influencing stochastic dynamics by gambling with limited resource, arXiv:2002.10172 [cs.AI], 2020.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 4.

The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265, A171890, A213652, A213651.
Columns for k=0..9 (with some shifts) are: A000012, A000027, A000217, A000292, A000332, A000389, A062989, A063262, A063263, A063264.
Cf. A000400, A008587, A018901, A063261, A063419.

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 6-nomials as a table
r := 6:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013

Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)

concat(vector(5,k,Vec(sum(j=0,5,x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

G.f. for row n: (Sum_{j=0..5} x^j)^n.
G.f. for column k: (x^(ceiling(k/5)))*N6(k, x)/(1-x)^(k+1) with the row polynomials from the staircase array A063261(k, m) and with N6(6,x) = 5 - 10*x + 10*x^2 - 5*x^3 + x^4.
T(n, k) = 0 if n=-1 or k<0 or k >= 5*n + 1; T(0, 0)=1; T(n, k) = Sum_{j=0..5} T(n-1, k-j) else.
T(n, k) = Sum_{i = 0..floor(k/6)} (-1)^i*binomial(n,i)*binomial(n+k-1-6*i,n-1) for n >= 0 and 0 <= k <= 5*n. - Peter Bala, Sep 07 2013
T(n, k) = Sum_{i = max(0,ceiling((k-2*n)/3)).. min(n,k/3)} binomial(n,i)*trinomial(n,k-3*i) for n >= 0 and 0 <= k <= 5*n. - Matthew Monaghan, Sep 30 2015

More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002

A063262 Eighth column (k=7) of sextinomial array A063260.

Original entry on oeis.org

4, 27, 104, 305, 756, 1667, 3368, 6354, 11340, 19327, 31680, 50219, 77324, 116055, 170288, 244868, 345780, 480339, 657400, 887589, 1183556, 1560251, 2035224, 2628950, 3365180, 4271319, 5378832

Offset: 0

Wolfdieter Lang, Jul 24 2001

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A062989, A063260, A063261, A063263, A063264, A213888.

CoefficientList[Series[(4-5x+5x^3-4x^4+x^5)/(1-x)^8,{x,0,30}],x] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{4,27,104,305,756,1667,3368,6354},30] (* Harvey P. Dale, Mar 07 2023 *)

a(n) = A063260(n+2, 7 )= (n+1)*(n+2)*(n^5+32*n^4+413*n^3+2722*n^2+9432*n+10080)/7!.
G.f.: (4-5*x+5*x^3-4*x^4+x^5)/(1-x)^8; the numerator polynomial is N6(7, x) from row n=7 of array A063261.
a(n) = 4*C(n+2,2) + 15*C(n+2,3) + 20*C(n+2,4) + 15*C(n+2,5) + 6*C(n+2,6) + C(n+2,7) (see comment in A213888). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A063263 Ninth column (k=8) of sextinomial array A063260.

Original entry on oeis.org

3, 27, 125, 420, 1161, 2807, 6147, 12465, 23760, 43032, 74646, 124787, 202020, 317970, 488138, 732870, 1078497, 1558665, 2215875, 3103254, 4286579, 5846577, 7881525, 10510175, 13875030, 18145998, 23524452

Offset: 0

Wolfdieter Lang, Jul 24 2001

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A062989, A063260, A063261, A063262, A063264, A213888.

a(n) = A063260(n+2, 8) = (n+1)*(n+2)*(n+3)*(n^5+38*n^4+587*n^3+4678*n^2+19896*n+20160)/8!.
G.f.: (3-10*x^2+15*x^3-9*x^4+2*x^5)/(1-x)^9; the numerator polynomial is N6(8, x) from row n=8 of array A063261.
a(n) = 3*C(n+2,2) + 18*C(n+2,3) + 35*C(n+2,4) + 35*C(n+2,5) + 21*C(n+2,6) + 7*C(n+2,7) + C(n+2,8) (see comment in A213888). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A063264 Tenth column (k=9) of sextinomial array A063260.

Original entry on oeis.org

2, 25, 140, 540, 1666, 4417, 10480, 22825, 46420, 89232, 163592, 288015, 489580, 806990, 1294448, 2026502, 3104030, 4661555, 6876100, 9977814, 14262622, 20107175, 27986400, 38493975, 52366080, 70508802

Offset: 0

Wolfdieter Lang, Jul 24 2001

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A062989, A063260, A063261, A063262, A063263, A213888.

a(n) = A063260(n+2, 9) = (n+1)*(n+2)*(n+3)*(n+4)*(n^5+44*n^4+791*n^3+7384*n^2+37140*n+30240)/9!.
G.f.: (2+5*x-20*x^2+25*x^3-14*x^4+3*x^5)/(1-x)^10; the numerator polynomial is N6(8, x) from row n=8 of array A063261.
a(n) = 2*C(n+2,2) + 19*C(n+2,3) + 52*C(n+2,4) + 70*C(n+2,5) + 56*C(n+2,6) + 28*C(n+2,7) + 8*C(n+2,8) + C(n+2,9) (see comment in A213888). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A213744 Triangle of numbers C^(5)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 5 appearances allowed.

Original entry on oeis.org

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, 456, 1, 7, 28, 84, 210, 462, 917, 1667, 1, 8, 36, 120, 330, 792, 1708, 3368, 6147, 1, 9, 45, 165, 495, 1287, 2994, 6354, 12465, 22825, 1, 10

Offset: 0

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

For k<=4, the triangle coincides with triangle A213743.

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)=A062989(n) for n>=5, T(n,7)=A063262 for n>=5, T(n,8)=A063263 for n>=6, T(n,9)=A063264 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...456
.7..|..1.....7....28....84...210....462...917....1667

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

Cf. A007318, A005725, A111808, A187925, A213742, A213743, A000217, A000292, A000332, A000389, A062989, A063262, A063263, A063264.

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}]&[6] (* Peter J. C. Moses, Apr 16 2013 *)

C^(5)(n,k)=sum{r=0,...,floor(k/6)}(-1)^r*C(n,r)*C(n-6*r+k-1, n-1)

A062990 Eighth column (r=7) of FS(5) staircase array A062985.

Original entry on oeis.org

5, 30, 110, 315, 771, 1688, 3396, 6390, 11385, 19382, 31746, 50297, 77415, 116160, 170408, 245004, 345933, 480510, 657590, 887799, 1183787, 1560504, 2035500, 2629250, 3365505, 4271670, 5379210, 6724085, 8347215

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 7}_{4}, n >= 0.

Harry J. Smith, Table of n, a(n) for n = 0..1000
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Partial sums of A062989.

[seq((binomial(n+7,n)-binomial(n+2,n)),n=1..29)]; # Zerinvary Lajos, Jun 23 2006

Table[Binomial[n+7,n]-Binomial[n+2,n],{n,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{5,30,110,315,771,1688,3396,6390},30] (* Harvey P. Dale, Jun 09 2016 *)

{ for (n=0, 1000, m=n + 1; a=binomial(m + 7, m) - binomial(m + 2, m); write("b062990.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 15 2009

a(n) = A062985(n+2, 7) = (n+1)*(n+2)*(n+3)*(n^4 + 29*n^3 + 326*n^2 + 1744*n + 4200)/7!.
G.f.: N(5;1, x)/(1-x)^8 with N(5;1, x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x polynomial of second row of A062986.
a(n) = binomial(n+7,n) - binomial(n+2,n). - Zerinvary Lajos, Jun 23 2006

More terms from Zerinvary Lajos, Jun 23 2006

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A063260 Sextinomial (also called hexanomial) coefficient array.

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A063262 Eighth column (k=7) of sextinomial array A063260.

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A063263 Ninth column (k=8) of sextinomial array A063260.

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A063264 Tenth column (k=9) of sextinomial array A063260.

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A213744 Triangle of numbers C^(5)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 5 appearances allowed.

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A062990 Eighth column (r=7) of FS(5) staircase array A062985.

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