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A154929
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A Fibonacci convolution triangle.
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10
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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)). - DELEHAM Philippe, Jan 25 2012
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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 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 18 2009]
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000045(n+1), A028859(n), A125145(n), A086347(n+1) for x=0,1,2,3 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 19 2009]
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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 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 18 2009]
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 - DELEHAM Philippe, Jan 25 2012
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CROSSREFS
| Sequence in context: A055888 A094442 A060642 * A049400 A106382 A004741
Adjacent sequences: A154926 A154927 A154928 * A154930 A154931 A154932
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Jan 17 2009
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