A121574 Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).

1, 2, 1, 4, 5, 1, 8, 16, 8, 1, 16, 44, 37, 11, 1, 32, 112, 134, 67, 14, 1, 64, 272, 424, 305, 106, 17, 1, 128, 640, 1232, 1168, 584, 154, 20, 1, 256, 1472, 3376, 3992, 2641, 998, 211, 23, 1, 512, 3328, 8864, 12592, 10442, 5221, 1574, 277, 26, 1

Offset: 0

Author

Paul Barry, Aug 08 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A006190(n+1); diagonal sums are A077939.

Inverse is A121575.

A generalized Delannoy number triangle.

Antidiagonal sums are A002478. - Philippe Deléham, Nov 10 2011.

From Peter Bala, Feb 07 2024: (Start)

The following remarks assume the row indexing starts at n = 1.

The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = 2 + x, R(3,x) = 4 + 5*x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)

Examples

			Triangle begins
   1;
   2,   1;
   4,   5,   1;
   8,  16,   8,   1;
  16,  44,  37,  11,   1;
  32, 112, 134,  67,  14,  1;
  64, 272, 424, 305, 106, 17, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• M. Norfleet, Characterization of second-order strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166-169.

Crossrefs

Cf. Diagonals: A000012, A016789, A080855, A000079, A053220.

Programs

GAP

T:=Flat(List([0..9],n->List([0..n],k->Sum([0..n-k],j->Binomial(k,j)*Binomial(n-j,k)*2^(n-k-j))))); # Muniru A Asiru, Nov 02 2018

Magma

[[(&+[ Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j): j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018

Maple

T:=(n,k)->add(binomial(k,j)*binomial(n-j,k)*2^(n-k-j),j=0..n-k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Nov 02 2018

Mathematica

Table[Sum[Binomial[k, j] Binomial[n-j, k] 2^(n-k-j), {j, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 02 2018 *)

PARI

for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, binomial(k, j)* binomial(n-j, k)*2^(n-k-j)), ", "))) \\ G. C. Greubel, Nov 02 2018

Formula

Number array T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-k-j).

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1). - Philippe Deléham, Nov 10 2011

Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = (x + 2)*R(n-1,x) + x*R(n-2,x) with R(1,x) = 1 and R(2,x) = x + 2. - Peter Bala, Feb 07 2024