A035343 -id:A035343 - cp's OEIS frontend

A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.

1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780, 780, 735, 651, 540, 420, 305, 205, 126, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 462, 786, 1251

Offset: 0

Matthew Vandermast, Oct 31 2010

In each row n>=0, k takes values from 0 to n^2 inclusive. Row sums equal A000169(n+1). All rows are palindromic. Row n is also row n of the (n+1)-nomial array (e.g., row 1 is also row 1 of A007318).
T(n,k) gives the number of divisors of A181555(n) with k prime factors counted with multiplicity. See also A001222, A071207, A146291, A146292.
T(n,k) is the number of size k submultisets of the so-called regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 0 < i < n). - Thomas Wieder, Dec 28 2013

			Rows begin:
1;
1,1;
1,2,3,2,1;
1,3,6,10,12,12,10,6,3,1;...
T(n=3,k=4) = 12 because we have 12 submultisets (without regard of the order of elements) of size k=4 for the regular multiset (n=3) {1, 1, 1, 2, 2, 2, 3, 3, 3}: {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 3, 3}, {1, 2, 2, 2}, {1, 2, 2, 3}, {1, 2, 3, 3}, {1, 3, 3, 3}, {2, 2, 2, 3}, {2, 2, 3, 3}, {2, 3, 3, 3}.

Alois P. Heinz, Rows n = 0..32, flattened
Anonymous, Polynomial calculator
Thomas Wieder, A181567 as Excel table
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

A163181 gives row n of n-nomial array. See also A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890.

b:= proc(n, k, i) option remember; `if`(k=0, 1,
     `if`(i<1, 0, add(b(n, k-j, i-1), j=0..n)))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n^2), n=0..8); # Alois P. Heinz, Jul 04 2016

row[n_] := CoefficientList[((x^(n+1) - 1)/(x-1))^n + O[x]^(n^2+1), x]; Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Apr 06 2017 *)

A000575 Tenth column of quintinomial coefficients.

Original entry on oeis.org

10, 80, 365, 1246, 3535, 8800, 19855, 41470, 81367, 151580, 270270, 464100, 771290, 1245488, 1960610, 3016820, 4547840, 6729800, 9791859, 14028850, 19816225, 27627600, 38055225, 51833730, 69867525, 93262260, 123360780, 161784040, 210477476, 271763360

Offset: 0

N. J. A. Sloane

In the Carlitz et al. reference a(n)= Q_{5,n+2}(2), n >= 0, with a(n)=binomial(11+n,n+2)-(n+3)*binomial(n+6,n+2), (eq.(3.3), p. 356, with n=5, m->n+2,r=2). Q_{5,m}(2) is the number of sequences (i_1,i_2,...,i_m) with i_s, s=1,...,m, from {1,2,3,4,5} (repetitions allowed), with exactly 2 increases between successive elements (first position is counted as an increase).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374, p. 351.
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

CoefficientList[Series[(10-20*x+15*x^2-4*x^3)/(1-x)^10,{x,0,50}],x](* Vincenzo Librandi, Mar 28 2012 *)

a(n) = polcoeff((1+x+x^2+x^3+x^4)^(n+3), 9); \\ Joerg Arndt, Aug 04 2015

a(n) = A035343(n+3, 9) = binomial(n+6, 6)*(n^3+42*n^2+677*n+5040)/(9!/6!).
G.f.: (10-20*x+15*x^2-4*x^3)/(1-x)^10; numerator polynomial is N5(9, x) from the array A063422.
a(n) = 10*C(n+3,3) + 40*C(n+3,4) + 65*C(n+3,5) + 56*C(n+3,6) + 28*C(n+3,7) + 8*C(n+3,8) + C(n+3,9) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = Sum_{k=1..10} (-1)^k * binomial(10,k) * a(n-k), a(0)=10. - G. C. Greubel, Aug 03 2015
a(n) = [x^9] (1+x+x^2+x^3+x^4)^(n+3). - Joerg Arndt, Aug 04 2015

Comments and more terms from Wolfdieter Lang, Aug 29 2001

More terms from Sean A. Irvine, Nov 24 2010

A213652 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 52, 57, 60, 61, 60, 57, 52, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456, 480, 489, 480, 456

Offset: 0

M. F. Hasler, Jun 17 2012

The n-th row also yields the number of ways to get a total of n, n+1,..., 9n, when summing n integers ranging from 1 to 9.
The row sums equal 9^n = A001019(n).
The row lengths are 1+8n = A017077(n).

			The triangle starts:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1; (row sum = 9, row length = 9)
(row n=2) 1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1; (sum = 81, length = 17)
(row n=3) 1,3,6,10,15,21,28,36,45,52,57,60,61,60,... (sum = 729, length = 25)
(row n=4) 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456,... (sum = 9^4; length = 33),
etc.

Seiichi Manyama, Rows n = 0..49, flattened

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.

Cf. A001019, A017077.

#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 9-nomials as a table
r := 9:  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

concat(vector(5,k,Vec(sum(j=0,8,x^j)^(k-1))))

T(n,k) = Sum_{i=0..floor(k/9)} (-1)^i*binomial(n,i)*binomial(n+k-1-9*i,n-1) for n >= 0 and 0 <= k <= 8*n. - Peter Bala, Sep 07 2013

A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10

Offset: 1

Andrey Zabolotskiy, Nov 10 2016

Equivalently, T(k,n,h) is the number of ordered sets of n nonnegative integers < k with the sum equal to h.
From Juan Pablo Herrera P., Nov 21 2016: (Start)
T(k,n,h) is the number of possible ways of randomly selecting h cards from k-1 sets, each with n different playing cards. It is also the number of lattice paths from (0,0) to (n,h) using steps (1,0), (1,1), (1,2), ..., (1,k-1).
Shallow diagonal sums of each triangle with fixed k give the k-bonacci numbers. (End)
T(k,n,h) is the number of n-dimensional grid points of a k X k X ... X k grid, which are lying in the (n-1)-dimensional hyperplane which is at an L1 distance of h from one of the grid's corners, and normal to the corresponding main diagonal of the grid. - Eitan Y. Levine, Apr 23 2023

			For first few k and for first few n, the rows with h = 0..n*(k-1) are given:
k=1:  1;  1;  1;  1;  1; ...
k=2:  1;  1, 1;  1, 2, 1;  1, 3, 3, 1;  1, 4, 6, 4, 1; ...
k=3:  1;  1, 1, 1;  1, 2, 3, 2, 1;  1, 3, 6, 7, 6, 3, 1; ...
k=4:  1;  1, 1, 1, 1;  1, 2, 3, 4, 3, 2, 1; ...
For example, (1 + x + x^2)^3 = 1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6, hence T(3,3,2) = T(3,3,4) = 6.
From _Eitan Y. Levine_, Apr 23 2023: (Start)
Example for the repeated cumulative sum formula, for (k,n)=(3,3) (each line is the cumulative sum of the previous line, and the first line is the padded, alternating 3rd row from Pascal's triangle):
  1  0  0 -3  0  0  3  0  0 -1
  1  1  1 -2 -2 -2  1  1  1
  1  2  3  1 -1 -3 -2 -1
  1  3  6  7  6  3  1
which is T(3,3,h). (End)

Andrey Zabolotskiy, slices with k+n = 1..20, flattened
Florentin Smarandache, K-Nomial Coefficients, arXiv:math/0612062 [math.GM], 2006 (originally published in French in: F. Smarandache, Généralisations et Généralités, Ed. Nouvelle, 1984, pp. 24-26).

k-nomial arrays for fixed k=1..10: A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.
Arrays for fixed n=0..6: A000012, A000012, A004737, A109439, A277949, A277950, A277951.
Central n-nomial coefficients for n=1..9, i.e., sequences with h=floor(n*(k-1)/2) and fixed n: A000012, A000984 (A001405), A002426, A005721 (A005190), A005191, A063419 (A018901), A025012, (A025013), A025014, A174061 (A025015), A201549, (A225779), A201550. Arrays: A201552, A077042, see also cfs. therein.
Triangle n=k-1: A181567. Triangle n=k: A163181.

a = Table[CoefficientList[Sum[x^(h-1),{h,k}]^n,x],{k,10},{n,0,9}];
Flatten@Table[a[[s-n,n+1]],{s,10},{n,0,s-1}]
(* alternate program *)
row[k_, n_] := Nest[Accumulate,Upsample[Table[((-1)^j)*Binomial[n,j],{j,0,n}],k],n][[;;n*(k-1)+1]] (* Eitan Y. Levine, Apr 23 2023 *)

T(k,n,h) = Sum_{i = 0..floor(h/k)} (-1)^i*binomial(n,i)*binomial(n+h-1-k*i,n-1). [Corrected by Eitan Y. Levine, Apr 23 2023]
From Eitan Y. Levine, Apr 23 2023: (Start)
(T(k,n,h))_{h=0..n*(k-1)} = f(f(...f(g(P))...)), where:
(x_i)_{i=0..m} denotes a tuple (in particular, the LHS contains the values for 0 <= h <= n*(k-1)),
f repeats n times,
f((x_i){i=0..m}) = (Sum{j=0..i} x_j)_{i=0..m} is the cumulative sum function,
g((x_i){i=0..m}) = (x(i/k) if k|i, otherwise 0)_{i=0..m*k} is adding k-1 zeros between adjacent elements,
and P=((-1)^i*binomial(n,i))_{i=0..n} is the n-th row of Pascal's triangle, with alternating signs. (End)
From Eitan Y. Levine, Jul 27 2023: (Start)
Recurrence relations, the first follows from the sequence's defining polynomial as mentioned in the Smarandache link:
T(k,n+1,h) = Sum_{i = 0..s-1} T(k,n,h-i)
T(k+1,n,h) = Sum_{i = 0..n} binomial(n,i)*T(k,n-i,h-i*k) (End)

A027659 a(n) = binomial(n+2,2) + binomial(n+3,3) + binomial(n+4,4) + binomial(n+5,5).

Original entry on oeis.org

4, 18, 52, 121, 246, 455, 784, 1278, 1992, 2992, 4356, 6175, 8554, 11613, 15488, 20332, 26316, 33630, 42484, 53109, 65758, 80707, 98256, 118730, 142480, 169884, 201348, 237307, 278226, 324601, 376960, 435864, 501908, 575722, 657972, 749361, 850630, 962559

Offset: 0

N. J. A. Sloane

Harvey P. Dale, 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. A035343, A062750, A063422.

Partial sums of A063258.

[Binomial(n+6, 5) -(n+2): n in [0..60]]; // G. C. Greubel, Aug 01 2022

seq(1/120*(n+8)*(n+2)*(n+1)*(n^2+9*n+30), n=0..40);

Table[Sum[Binomial[n+i,i],{i,2,5}],{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20, -15,6,-1}, {4,18,52,121,246,455},30] (* Harvey P. Dale, Aug 18 2012 *)
Sum[(-1)^j*Binomial[4*j-2 + Range[0, 60], 4*j-3], {j,2}] (* G. C. Greubel, Aug 01 2022 *)

a(n)=(n+8)*(n+2)*(n+1)*(n^2+9*n+30)/120 \\ Charles R Greathouse IV, Oct 07 2015

[binomial(n+6, 5) -(n+2) for n in (0..60)] # G. C. Greubel, Aug 01 2022

a(n) = A035343(n+2, 5), n >= 0 (sixth column of quintinomial coefficients).
a(n) = A062750(n+2, 5), n >= 0 (sixth column).
G.f.: (x^2)*(2-x)*(2 - 2*x + x^2)/(1-x)^6. (For numerator polynomial see N5(5, x) = 4 - 6*x + 4*x^2 - x^3 from A063422.)
a(n) = binomial(n+6, 5) - binomial(n+2, 1). - Zerinvary Lajos, May 08 2006
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), with a(0)=4, a(1)=18, a(2)=52, a(3)=121, a(4)=246, a(5)=455. - Harvey P. Dale, Aug 18 2012
From G. C. Greubel, Aug 01 2022: (Start)
a(n) = Sum_{j=0..3} binomial(n+j+2, j+2).
E.g.f.: (1/120)*(480 +1680*x +1200*x^2 +300*x^3 +30*x^4 +x^5)*exp(x). (End)

A063422 Coefficient array for certain numerator polynomials N5(n,x), n >= 0 (rising powers of x) used for quintinomials (also called pentanomials).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, -6, 4, -1, 3, -2, -2, 3, -1, 2, 2, -8, 7, -2, 1, 6, -14, 11, -3, 10, -20, 15, -4, 6, 2, -37, 65, -56, 28, -8, 1, 3, 16, -61, 78, -42, 0, 12, -6, 1, 1, 22, -57, 35, 42, -84, 60, -21, 3, 20, -25, -64, 196, -224, 136, -44, 6, 10, 35, -219, 420

Offset: 0

Wolfdieter Lang, Jul 27 2001

The g.f. of column k of array A035343(n,k) (quintinomial coefficients) is (x^(ceiling(k/4)))*N5(k,x)/(1-x)^(k+1).
The sequence of degrees for the polynomials N5(n,x) is [0, 0, 0, 0, 0, 3, 4, 4, 4, 3, 7, 8, 8, 7, 7,...] for n >= 0.
Row sums N5(n,1)=1 for all n.

			{1}; {1}; {1}; {1}; {1}; {4,-6,4,-1}; {3,-2,-2,3,-1}; {2,2,-8,7,-2}; {1,6,-14,11,-3}; ...
c=2: b(2,1)= 1 = b(2,2), b(2,3)= 0 =b(2,4).
N5(6,x)=3-2*x-2*x^2+3*x^3-x^4.

a(n, m) = [x^m]N5(n, x), n, m >= 0, with N5(n, x)= sum(((1-x)^(j-1))*(x^(b(c(n), j)))*N5(n-j, x), j=1..4), N5(n, x)= 1 for n=0..3 and b(c(n), j) := 1 if 1<= j <= c(n) else 0, with c(n) := 3 if mod(n, 4)=0 else c(n) := mod(n, 4)-1; (hence b(0, j)=0, j=1..4).

A064056 Seventh column of quintinomial coefficients.

Original entry on oeis.org

3, 19, 68, 185, 426, 875, 1652, 2922, 4905, 7887, 12232, 18395, 26936, 38535, 54008, 74324, 100623, 134235, 176700, 229789, 295526, 376211, 474444, 593150, 735605, 905463, 1106784, 1344063, 1622260, 1946831

Offset: 0

Wolfdieter Lang, Aug 29 2001

Cf. A027659 (sixth column).

a(n) = A035343(n+2, 6) = binomial(n+2, 2)*(n^4+24*n^3+221*n^2+954*n+1080)/(6!/2!), n >= 0.
G.f.: (3-2*x-2*x^2+3*x^3-x^4)/(1-x)^7; numerator polynomial is N5(6, x) from the array A063422.
a(n) = 3*C(n+2,2) + 10*C(n+2,3) + 10*C(n+2,4) + 5*C(n+2,5) + C(n+2,6) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A064057 Eighth column of quintinomial coefficients.

Original entry on oeis.org

2, 18, 80, 255, 666, 1520, 3144, 6030, 10890, 18722, 30888, 49205, 76050, 114480, 168368, 242556, 343026, 477090, 653600, 883179, 1178474, 1554432, 2028600, 2621450, 3356730, 4261842, 5368248

Offset: 0

Wolfdieter Lang, Aug 29 2001

Cf. A064056 (seventh column).

a(n) = A035343(n+2, 7)= binomial(n+3, 3)*(n+14)*(n^3+15*n^2+116*n+120)/(7!/3!).
G.f.: (2+2*x-8*x^2+7*x^3-2*x^4 )/(1-x)^8; numerator polynomial is N5(7, x) from the array A063422.
a(n) = 2*C(n+2,2) + 12*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 A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A163181 T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 456, 756, 1161, 1666, 2247, 2856, 3431, 3906, 4221, 4332, 4221, 3906, 3431

Offset: 1

Geoffrey Critzer, Jul 22 2009

T(n,k) is the number of length n sequences on an alphabet of {0,1,2,...,n-1} that have a sum of k. Equivalently T(n,k) is the number of functions f:{1,2,...,n}->{0,1,2,...,n-1} such that Sum(f(i)=k, i=1...n).

Row n is also row n of the array of q-nomial coefficients. - Matthew Vandermast, Oct 31 2010

			T(3,4) = 6 because there are 6 ternary sequences of length three that sum to 4: [0, 2, 2], [1, 1, 2], [1, 2, 1], [2, 0, 2], [2, 1, 1], [2, 2, 0].

Alois P. Heinz, Rows n = 1..32, flattened

The maximum of row n is in column k=n(n-1)/2 = A000217(n-1).

For q-nomial arrays, see A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890. See also A181567. - Matthew Vandermast, Oct 31 2010

b:= proc(n, k, l) option remember; `if`(k=0, 1,
      `if`(l=0, 0, add(b(n, k-j, l-1), j=0..min(n-1, k))))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n*(n-1)), n=1..8);  # Alois P. Heinz, Feb 21 2013

(*warning very inefficient*) Table[Distribution[Map[Total, Strings[Range[n], n]]], {n, 1, 6}]//Grid
nn=100;Table[CoefficientList[Series[Sum[x^i,{i,0,n-1}]^n,{x,0,nn}],x],{n,1,10}]//Grid (* Geoffrey Critzer, Feb 21 2013*)

O.g.f. for row n is ((1-x^n)/(1-x))^n. For k<=(n-1), T(n,k) = C(n+k-1,k).

A308087 Number of lattice paths from (0,0) to (n,n) using Euclid's orchard as a step-set.

Original entry on oeis.org

1, 1, 1, 3, 13, 45, 153, 515, 1767, 6167, 21697, 76661, 271973, 968561, 3460677, 12399661, 44534647, 160285049, 577949447, 2087375443, 7550053527, 27344761057, 99155777619, 359943568005, 1307923066305, 4756914915657, 17315390737219, 63077564876055

Offset: 0

Nicholas Ham, May 11 2019

Alois P. Heinz, Table of n, a(n) for n = 0..575
J. East and N. C. Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
N. C. Ham, Implementation of algorithms 1-3 from J. East and N. C. Ham reference above.
Wikipedia, Euclid's orchard

Cf. A001764, A005043, A005165, A035343, A067955, A097609, A308112, A308113.

b:= proc(x, y) option remember; `if`(y=0, 1, add(add(`if`(1=
      igcd(h, v), b(sort([x-h, y-v])[]), 0), v=1..y), h=1..x))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 12 2019

b[x_, y_] := b[x, y] = If[y == 0, 1, Sum[Sum[If[1 == GCD[h, v], b @@ Sort[{x - h, y - v}], 0], {v, 1, y}], {h, 1, x}]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Feb 29 2020, after Alois P. Heinz *)

a(n) mod 2 = 1. - Alois P. Heinz, May 13 2019

a(n) ~ c * d^n / sqrt(n), where d = 3.7137893481485186502229788321701955452444... and c = 0.133597878112414800677299372849715598093... - Vaclav Kotesovec, May 24 2019

a(16)-a(27) from Alois P. Heinz, May 12 2019

cp's OEIS Frontend

A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.

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A000575 Tenth column of quintinomial coefficients.

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A213652 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...

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A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

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A027659 a(n) = binomial(n+2,2) + binomial(n+3,3) + binomial(n+4,4) + binomial(n+5,5).

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A063422 Coefficient array for certain numerator polynomials N5(n,x), n >= 0 (rising powers of x) used for quintinomials (also called pentanomials).

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A064056 Seventh column of quintinomial coefficients.

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A064057 Eighth column of quintinomial coefficients.

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A163181 T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).