A154929 A Fibonacci convolution triangle.

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 8, 22, 21, 8, 1, 13, 45, 59, 36, 10, 1, 21, 88, 147, 124, 55, 12, 1, 34, 167, 339, 366, 225, 78, 14, 1, 55, 310, 741, 976, 770, 370, 105, 16, 1, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 144, 1020, 3174, 5696, 6505, 4920, 2485

Offset: 0

Author

Paul Barry, Jan 17 2009

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A028859. Diagonal sums are A141015(n+1). Inverse is A154930. Product of A030528 and A007318.

Transforms sequence m^n with g.f. 1/(1-m*x) to the sequence with g.f. (1+x)/(1-(m+1)x-(m+1)x^2).

Subtriangle of triangle T(n,k), given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. This triangle is the Riordan array (1, x(1+x)/(1-x-x^2)). - Philippe Deléham, Jan 25 2012

Examples

			Triangle begins
   1;
   2,   1;
   3,   4,   1;
   5,  10,   6,   1;
   8,  22,  21,   8,   1;
  13,  45,  59,  36,  10,   1;
  21,  88, 147, 124,  55,  12,   1;
  34, 167, 339, 366, 225,  78,  14,  1;
  55, 310, 741, 976, 770, 370, 105, 16, 1;
Production array is
     2,    1;
    -1,    2,   1;
     3,   -1,   2,   1;
   -10,    3,  -1,   2,  1;
    36,  -10,   3,  -1,  2,  1;
  -137,   36, -10,   3, -1,  2, 1;
   543, -137,  36, -10,  3, -1, 2, 1;
or ((1+x+sqrt(1+6x+5x^2))/2,x) beheaded.
T(5,3) = T(4,3) + T(4,2) + T(3,3) + T(3,2) = 8 + 21 + 1 + 6 = 36. - _Philippe Deléham_, Jan 18 2009
From _Philippe Deléham_, Jan 25 2012: (Start)
Triangle (0,2,-1/2,-1/2,0,0,0,...) DELTA (1,0,0,0,0,0,...) begins:
  1;
  0,   1;
  0,   2,   1;
  0,   3,   4,   1;
  0,   5,  10,   6,   1;
  0,   8,  22,  21,   8,   1;
  0,  13,  45,  59,  36,  10,   1;
  0,  21,  88, 147, 124,  55,  12,   1; (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
• Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Programs

Mathematica

Table[Sum[Binomial[j + 1, n - j] Binomial[j, k], {j, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Formula

Riordan array ((1+x)/(1-x-x^2), x(1+x)/(1-x-x^2));

Triangle T(n,k) = Sum_{j=0..n} C(j+1,n-j)*C(j,k).

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(n,k)=0 if k > n. - Philippe Deléham, Jan 18 2009

Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A028859(n), A125145(n), A086347(n+1) for x=0,1,2,3 respectively. - Philippe Deléham, Jan 19 2009