A008287 -id:A008287 - cp's OEIS frontend

A171890 Octonomial coefficient array.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35

Offset: 0

N. J. A. Sloane, Oct 19 2010

Row lengths are 1,8,15,22,... = 1+7n = A016993(n). Row sums are 1,8,64,... = 8^n = A001018(n). M. F. Hasler, Jun 17 2012

			Array begins:
[1]
[1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1]
...

T. D. Noe, Rows n = 0..25, flattened

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

#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 8-nomials as a table
r := 8:  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 + x^6 + x^7)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *)

concat(vector(5, k, Vec(sum(j=0, 7, x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

Row n has g.f. (1+x+...+x^7)^n.

T(n,k) = sum {i = 0..floor(k/8)} (-1)^i*binomial(n,i)*binomial(n+k-1-8*i,n-1) for n >= 0 and 0 <= k <= 7*n. - Peter Bala, Sep 07 2013

A213651 10-nomial coefficient array: Coefficients of the polynomial (1 + ... + X^9)^n, n=0,1,...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 63, 69, 73, 75, 75, 73, 69, 63, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 282, 348, 415, 480

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, ..., 10n, when throwing n 10-sided dice, or summing n integers ranging from 1 to 10.
The row sums equal 10^n = A011557(n).
The row lengths are 1 + 9n = 10n - (n-1) = A017173(n).
T(n,k) is the number of integers in the [0, 10^n-1] range distributed according to the sum k of their digits. - Miquel Cerda, Jun 21 2017
The sum of the squares of the integers of the n-th row gives A174061(n). - Miquel Cerda, Jul 03 2017

			There are 1, 3, 6, 10, ... ways to score a total of 4, 5, 6, 7, ... when throwing three 10-sided dice.
The table begins as follows:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1,1; (row sum = 10, row length = 10)
(row n=2) 1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1; (sum = 100, length = 19)
(row n=3) 1,3,6,10,15,21,28,36,45,55,63,69,73,75,75,73,...; row sum = 1000;
(row n=4) 1,4,10,20,35,56,84,120,165,220,282,348,415,...; row sum = 10^4;
etc.
Number of integers in (row n=2): k(2)=3, because in the range 0 to 99 there are 3 integers whose digits sum to 2: 2, 11 and 20. - _Miquel Cerda_, Jun 21 2017

Seiichi Manyama, Rows n = 0..46, flattened
Miquel Cerda, Graphical construction of the triangle T(n,k) for n = 0..11.

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

#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 10-nomials as a table
r := 10:  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=0,9,x^j)^(k-1))))

T(n,k) = Sum_{i = 0..floor(k/10)} (-1)^i*binomial(n,i)*binomial(n+k-1-10*i,n-1) for n >= 0 and 0 <= k <= 9*n. - Peter Bala, Sep 07 2013

A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).

Original entry on oeis.org

3, 12, 31, 65, 120, 203, 322, 486, 705, 990, 1353, 1807, 2366, 3045, 3860, 4828, 5967, 7296, 8835, 10605, 12628, 14927, 17526, 20450, 23725, 27378, 31437, 35931, 40890, 46345, 52328, 58872, 66011, 73780, 82215, 91353, 101232, 111891, 123370, 135710, 148953, 163142, 178321, 194535

Offset: 0

N. J. A. Sloane

If Y is an (n-3)-subset of an n-set X then, for n>=5, a(n-5) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
This equation represents the number of numbers with <=n digits such that the sum of the digits is between 1 and 4 inclusive and no digit is larger than 3. - David Consiglio, Jr., Oct 27 2008
Row 2 of the convolution array A213548. - Clark Kimberling, Jun 20 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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
Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, Bootstrapping cascaded random matrix models: correlations in permutations of matrix products, arXiv:2405.02541 [math-ph], 2024. See p. 7.
R. K. Guy, Letter to N. J. A. Sloane, 1987.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008287, A062745, A063421, A071675, A213548.

[(((n+14)*n+71)*n+130)*n/24+3: n in [0..45]]; // Vincenzo Librandi, Jun 15 2011

A005718:=-(3-3*z+z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Table[Plus@@Table[Binomial[i + n, n], {i, 2, 4}], {n, 0, 43}] (* From Alonso del Arte, Jun 14 2011 *)

a(n)=(((n+14)*n+71)*n+130)*n/24+3 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = binomial(n, 2)*(n^2+7*n+18)/12, n >= 2.
G.f.: (3-3*x+x^2)/(1-x)^5. (numerator polynomial is N4(4, x) from A063421).
a(n) = A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).
a(n) = A062745(n, 4), n >= 2 (fifth column).
a(n) = 3*C(n+2,2) + 3*C(n+2,3) + C(n+2,4) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
E.g.f.: exp(x)*(72 + 216*x + 120*x^2 + 20*x^3 + x^4)/24. - Stefano Spezia, May 09 2024

Better description from Zerinvary Lajos, Dec 02 2005

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

Original entry on oeis.org

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 *)

A001919 Eighth column of quadrinomial coefficients.

Original entry on oeis.org

6, 40, 155, 456, 1128, 2472, 4950, 9240, 16302, 27456, 44473, 69680, 106080, 157488, 228684, 325584, 455430, 627000, 850839, 1139512, 1507880, 1973400, 2556450, 3280680, 4173390, 5265936, 6594165, 8198880, 10126336, 12428768, 15164952, 18400800, 22209990

Offset: 3

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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).

T. D. Noe, Table of n, a(n) for n = 3..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

seq(n*(n^2-1)*(n^2-4)*(n^2+21*n+180)/5040,n=3..34); # Emeric Deutsch, Jan 27 2005
A001919:=(3*z**2-8*z+6)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n*(n^2 - 1)*(n^2 - 4)*(n^2 + 21*n + 180)/5040, {n, 3, 50}] (* T. D. Noe, Aug 17 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{6,40,155,456,1128,2472,4950,9240},40] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = A008287(n, 7) = binomial(n+2, 5)*(n^2+21*n+180 )/42, n >= 3.
G.f.: (x^3)*(6-8*x+3*x^2 )/(1-x)^8. Numerator polynomial is N4(7, x) from array A063421.
a(n) = n(n^2-1)(n^2-4)(n^2+21n+180)/5040. - Emeric Deutsch, Jan 27 2005
a(n) = 6*C(n,3) + 16*C(n,4) + 15*C(n,5) + 6*C(n,6) + C(n,7) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(3)=6, a(4)=40, a(5)=155, a(6)=456, a(7)=1128, a(8)=2472, a(9)=4950, a(10)=9240, 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). - Harvey P. Dale, Mar 27 2013

More terms from Emeric Deutsch, Jan 27 2005

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)

A063421 Coefficient array for certain numerator polynomials N4(n,x), n >= 0 (rising powers of x) used for quadrinomials.

Original entry on oeis.org

1, 1, 1, 1, 3, -3, 1, 2, 0, -2, 1, 1, 3, -5, 2, 6, -8, 3, 3, 4, -16, 15, -6, 1, 1, 10, -20, 10, 3, -4, 1, 10, -9, -15, 27, -15, 3, 4, 17, -60, 66, -32, 6, 1, 22, -41, -6, 71, -74, 36, -9, 1, 15, 6, -105, 168, -111, 24, 9, -6, 1, 5, 45, -147, 133, 21

Offset: 0

Wolfdieter Lang, Jul 27 2001

The g.f. of column k of array A008287(n,k) (quadrinomial coefficients) is (x^(ceiling(k/3)))*N4(k,x)/(1-x)^(k+1).
The sequence of degrees for the polynomials N4(n,x) is [0, 0, 0, 0, 2, 3, 3, 2, 5, 6, 5, 5, 8, 8, 8,...] for n >= 0.
Row sums N4(n,1)=1 for all n.

			The irregular triangle begins:
  1;
  1;
  1;
  1;
  3, -3,  1;
  2,  0, -2, 1;
  1,  3, -5, 2;
  6, -8,  3;
  ...
For c=1: b(1,1) = 1, b(1,2) = 0 = b(1,3), and N4(6,x)=1+3*x-5*x^2+2*x^3.

Cf. A008287.

a(n, m) = [x^m] N4(n, x), n, m >= 0, with N4(n, x) = Sum_{j=1..3} ((1-x)^(j-1))*(x^(b(c(n), j)))*N4(n-j, x), N4(n, x) = 1 for n = 0, 1, 2 and b(c(n), j) := 1 if 1<= j <= c(n) else 0, with c(n) := 2 if mod(n, 3) = 0 else c(n) := mod(n, 3) - 1; (hence b(0, j) = 0, j=1..3).

A079474 Triangular array: for s=0 to r-1, a(r,s) = p(s)^(r-s), where p(s) is the s-th primorial number. (p(0)=1, p(1)=2, p(2)=23, p(3)=23*5,...).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 8, 36, 30, 1, 16, 216, 900, 210, 1, 32, 1296, 27000, 44100, 2310, 1, 64, 7776, 810000, 9261000, 5336100, 30030, 1, 128, 46656, 24300000, 1944810000, 12326391000, 901800900, 510510, 1, 256, 279936, 729000000, 408410100000

Offset: 1

Alford Arnold, Jan 15 2003

In the expansion of [1+x+x^2+...+x^(r-s)]^s, the x^n coefficient states how many factors of a(r,s) have n prime factors.

As a square array A(n,k) n>=0 k>=1 read by descending antidiagonals, A(n,k) when n>=1 is the least common period over the positive integers of the occurrence of the first n prime numbers as the k-th least operand in the respective integers' prime factorizations (written without exponents). - Peter Munn, Jan 25 2017

			Triangle starts
  1;
  1,  2;
  1,  4,    6;
  1,  8,   36,    30;
  1, 16,  216,   900,   210;
  1, 32, 1296, 27000, 44100, 2310;
  ...

Cf. A002110, A007318, A008287, A027907, A036035, A061742, A074139.

a(2n,n) gives A181555.

p:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*p(n-1)) end:
a:= (r, s)-> p(s)^(r-s):
seq(seq(a(r, s), s=0..r-1), r=0..10);  # Alois P. Heinz, Aug 22 2019

p[0] = 1; p[s_] := p[s] = Prime[s] p[s-1];
a[r_, s_] := p[s]^(r-s);
Table[a[r, s], {r, 0, 10}, {s, 0, r-1}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Edited by Don Reble, Nov 02 2005

A134660 Number of odd coefficients in (1 + x + x^2 + x^3)^n.

Original entry on oeis.org

1, 4, 4, 4, 4, 16, 4, 8, 4, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 4, 8, 4, 16, 16, 8, 8, 32, 16, 32, 4, 16, 16, 16, 16, 64, 16, 32, 16, 64, 64, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 8, 16, 8, 32, 32, 16, 16, 64, 32, 64, 4, 16, 16, 16, 16, 64, 16, 32, 16, 64, 64, 16, 16, 64

Offset: 0

Steven Finch, Jan 25 2008

nonn

			From _Omar E. Pol_, Mar 01 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
4;
4,4;
4,16,4,8;
4,16,16,4,4,16,8,16;
4,16,16,16,16,64,4,8,4,16,16,8,8,32,16,32;
4,16,16,16,16,64,16,32,16,64,64,4,4,16,8,16,4,16,16,16,16,64,8,16,8,32,32,16,16,64,32,64;
...
(End)

G. C. Greubel, Table of n, a(n) for n = 0..10000
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

Cf. A000120, A001316, A008287, A036555, A071053.

seq(igcd(4^n,binomial(4*n,n)),n=0..77); # Peter Luschny, Nov 08 2011

PolynomialMod[(1+x+x^2+x^3)^n, 2] /. x->1
A036555 = Total /@ IntegerDigits[3 Range[0, 100], 2]; Table[2^A036555[[n]], {n, 1, 20}] (* or *) Table[GCD[4^n, Binomial[4*n, n]], {n, 0, 50}] (* G. C. Greubel, Dec 31 2017 *)

a(n) = {my(pol= Pol([1,1,1,1], xx)*Mod(1,2)); subst(lift(pol^n), xx, 1);} \\ Michel Marcus, Mar 01 2015

a(n) = 2^hammingweight(3*n); \\ Joerg Arndt, Mar 10 2015

a(n) = 2^A036555(n).

a(n) = gcd(4^n, C(4*n, n)). - Peter Luschny, Nov 08 2011

cp's OEIS Frontend

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A079474 Triangular array: for s=0 to r-1, a(r,s) = p(s)^(r-s), where p(s) is the s-th primorial number. (p(0)=1, p(1)=2, p(2)=23, p(3)=23*5,...).