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A154929
A Fibonacci convolution triangle.
11
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
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Jan 17 2009
STATUS
approved