A104580 -id:A104580 - cp's OEIS frontend

A187889 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2+x^3)/(1-x-x^2-x^3)).

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 7, 19, 18, 7, 1, 13, 43, 54, 32, 9, 1, 24, 94, 147, 117, 50, 11, 1, 44, 200, 375, 375, 216, 72, 13, 1, 81, 418, 913, 1100, 799, 359, 98, 15, 1, 149, 861, 2147, 3027, 2657, 1507, 554, 128, 17, 1, 274, 1753, 4914, 7937, 8174, 5610, 2603, 809, 162, 19, 1

Offset: 0

Emanuele Munarini, Mar 15 2011

			Triangle begins:
1
1,1
2,3,1
4,8,5,1
7,19,18,7,1
13,43,54,32,9,1
24,94,147,117,50,11,1
44,200,375,375,216,72,13,1
81,418,913,1100,799,359,98,15,1

Cf. A104580, A321620.

(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1/(1 - x - x^2- x^3), 11] // Flatten (* Peter Luschny, Nov 27 2018 *)

trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
create_list(sum(binomial(i+k,k)*trinomial(i+k,n-k-i),i,0,n-k),n,0,8,k,0,n);

a(n,k) = Sum_{i=0..n-k} binomial(i+k,k)*trinomial(i+k,n-k-i), where trinomial(n,k) are the trinomial coefficients (A027907).

Recurrence: a(n+3,k+1) = a(n+2,k+1) + a(n+2,k) + a(n+1,k+1) + a(n+1,k) + a(n,k+1) + a(n,k)

A202193 Triangle read by rows: T(n,m) = coefficient of x^n in expansion of (x/(1 - x - x^2 - x^3 - x^4))^m.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 12, 9, 4, 1, 15, 28, 25, 14, 5, 1, 29, 62, 66, 44, 20, 6, 1, 56, 136, 165, 129, 70, 27, 7, 1, 108, 294, 401, 356, 225, 104, 35, 8, 1, 208, 628, 951, 944, 676, 363, 147, 44, 9, 1, 401, 1328, 2211, 2424, 1935, 1176, 553, 200, 54, 10, 1

Offset: 1

Vladimir Kruchinin, Dec 14 2011

From Philippe Deléham, Feb 16 2014: (Start)
As a Riordan array, this is (1/(1 - x - x^2 - x^3 - x^4), x/(1 - x - x^2 - x^3 - x^4)).
T(n,0) = A000078(n+3); T(n+1,1) = A118898(n+4).
Row sums are A103142(n).
Diagonal sums are A077926(n)*(-1)^n.
Tetranacci convolution triangle. (End)

			Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   4,  5,  3,  1;
   8, 12,  9,  4,  1;
  15, 28, 25, 14,  5,  1;
  29, 62, 66, 44, 20,  6,  1;

Similar sequences : A037027 (Fibonacci convolution triangle), A104580 (tribonacci convolution triangle). - Philippe Deléham, Feb 16 2014

T(n,m):=if n=m then 1 else sum(sum((-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1),i,0,(n-m-k)/4)*binomial(k+m-1,m-1),k,1,n-m);

T(n,m) = Sum_{k=1..n-m} (Sum_{i=0..floor((n-m-k)/4)} (-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1))*binomial(k+m-1,m-1), n > m, T(n,n)=1.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-3,k) + T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 16 2014
G.f. for column m: (x/(1 - x - x^2 - x^3 - x^4))^m. - Jason Yuen, Feb 17 2025

A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 7, 5, 1, 7, 17, 16, 7, 1, 13, 38, 46, 29, 9, 1, 24, 82, 122, 99, 46, 11, 1, 44, 174, 304, 303, 184, 67, 13, 1, 81, 362, 728, 857, 641, 309, 92, 15, 1, 149, 743, 1690, 2291, 2031, 1212, 482, 121, 17, 1, 274, 1509, 3827, 5869, 6004, 4260, 2108, 711, 154, 19, 1

Offset: 0

Emanuele Munarini, Apr 18 2011

Row sums are A077936, diagonal sums are A077946

			Triangle begins:
1
1,1
2,3,1
4,7,5,1
7,17,16,7,1
13,38,46,29,9,1
24,82,122,99,46,11,1
44,174,304,303,184,67,13,1
81,362,728,857,641,309,92,15,1

Cf. A104580, A187889, A077936, A077946.

Flatten[Table[Sum[Binomial[i+k,k]Sum[Binomial[i+k,j]Binomial[n-i-j,i+k],{j,0,n-k-2i}],{i,0,n}],{n,0,20},{k,0,n}]]

create_list(sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j,0,n-k-2*i),i,0,n),n,0,8,k,0,n);

T(n,k) = [x^n](x+x^2)^k/(1-x-x^2-x^3)^(k+1).
T(n,k) = sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j=0..n-k-2*i),i=0..n).
T(n,k) = sum(binomial(k,i)*(-1)^(k-i)*sum(binomial(j+k,k)*trinomial(i+j,n-3*k+2*i-j),j=0..n-k),i=0..k)
Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) + T(n+1,k) + T(n,k+1)

a(23) and a(40) corrected by Georg Fischer, Feb 20 2021 and Apr 29 2022

cp's OEIS Frontend

A187889 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2+x^3)/(1-x-x^2-x^3)).

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A202193 Triangle read by rows: T(n,m) = coefficient of x^n in expansion of (x/(1 - x - x^2 - x^3 - x^4))^m.

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A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).

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