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

0,2

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

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.

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

Example

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)

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 *)

Crossrefs

Sequence in context: A094442 A060642 A306186 * A249042 A262472 A049400

Adjacent sequences: A154926 A154927 A154928 * A154930 A154931 A154932

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 17 2009

Status

approved