A154929 - OEIS

%I #17 Jan 26 2020 20:56:51

%S 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,

%T 12,1,34,167,339,366,225,78,14,1,55,310,741,976,770,370,105,16,1,89,

%U 566,1557,2422,2337,1443,567,136,18,1,144,1020,3174,5696,6505,4920,2485

%N A Fibonacci convolution triangle.

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

%C 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).

%C 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

%H Michael De Vlieger, <a href="/A154929/b154929.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150)

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.

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

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

%F 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

%F 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

%e Triangle begins

%e    1;

%e    2,   1;

%e    3,   4,   1;

%e    5,  10,   6,   1;

%e    8,  22,  21,   8,   1;

%e   13,  45,  59,  36,  10,   1;

%e   21,  88, 147, 124,  55,  12,   1;

%e   34, 167, 339, 366, 225,  78,  14,  1;

%e   55, 310, 741, 976, 770, 370, 105, 16, 1;

%e Production array is

%e      2,    1;

%e     -1,    2,   1;

%e      3,   -1,   2,   1;

%e    -10,    3,  -1,   2,  1;

%e     36,  -10,   3,  -1,  2,  1;

%e   -137,   36, -10,   3, -1,  2, 1;

%e    543, -137,  36, -10,  3, -1, 2, 1;

%e or ((1+x+sqrt(1+6x+5x^2))/2,x) beheaded.

%e 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

%e From _Philippe Deléham_, Jan 25 2012: (Start)

%e Triangle (0,2,-1/2,-1/2,0,0,0,...) DELTA (1,0,0,0,0,0,...) begins:

%e   1;

%e   0,   1;

%e   0,   2,   1;

%e   0,   3,   4,   1;

%e   0,   5,  10,   6,   1;

%e   0,   8,  22,  21,   8,   1;

%e   0,  13,  45,  59,  36,  10,   1;

%e   0,  21,  88, 147, 124,  55,  12,   1; (End)

%t 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 *)

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Jan 17 2009