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A154929
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A Fibonacci convolution triangle.
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11
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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CROSSREFS
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Sequence in context: A094442 A060642 A306186 * A249042 A262472 A049400
Adjacent sequences: A154926 A154927 A154928 * A154930 A154931 A154932
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry, Jan 17 2009
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STATUS
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approved
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