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Search: id:A097094
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| A097094 |
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Triangle, read by rows, where the self-convolution of the antidiagonal sums (A097097) equals the row sums (A097096); the row sums also equals the convolution of the main diagonal (A097095) with the Fibonacci sequence (A000045). |
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+0
4
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| 1, 1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 4, 5, 5, 7, 1, 5, 8, 8, 10, 8, 1, 6, 12, 13, 15, 15, 15, 1, 7, 17, 21, 23, 25, 23, 19, 1, 8, 23, 33, 36, 40, 38, 34, 37, 1, 9, 30, 50, 57, 63, 63, 57, 56, 42, 1, 10, 38, 73, 90, 99, 103, 95, 90, 79, 66, 1, 11, 47, 103, 140, 156, 166, 158, 147, 135, 108
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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First negative term occurs in the 17th row.
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FORMULA
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T(n, 0) = 1 and T(n, n) = A097095(n) for n>=0; T(n, k) = T(n-1, k) + T(n-2, k-1) for n>k>=1. G.f.: A(x, y) = A097095(x*y)/(1-x-x^2*y), where A097095(x)/(1-x-x^2) = A097095(x^2)^2/(1-x-x^3)^2, and A097095(x) is the g.f. of the main diagonal.
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EXAMPLE
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Rows begin:
[1],
[1,1],
[1,2,2],
[1,3,3,3],
[1,4,5,5,7],
[1,5,8,8,10,8],
[1,6,12,13,15,15,15],
[1,7,17,21,23,25,23,19],
[1,8,23,33,36,40,38,34,37],
[1,9,30,50,57,63,63,57,56,42],
[1,10,38,73,90,99,103,95,90,79,66],
[1,11,47,103,140,156,166,158,147,135,108,76],
[1,12,57,141,213,246,265,261,242,225,187,142,122],...
Row sums are: {1,2,5,10,22,40,77,136,250,428,744,...}
and form the self-convolution of the antidiagonal sums, which begin: {1,1,2,3,6,8,14,20,35,49,77,112,176,...}.
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PROGRAM
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(PARI) {T(n, k)=if(n<k|k<0, 0, if(k==0|n<=1, 1, if(n>k, T(n-1, k)+T(n-2, k-1), polcoeff(Ser(vector(n+1, j, sum(i=0, j-1, T(j-i-1, i))))^2+x*O(x^n), n) -sum(i=0, n-1, T(n, i)) )))}
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CROSSREFS
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Cf. A097095, A097096, A097097.
Adjacent sequences: A097091 A097092 A097093 this_sequence A097095 A097096 A097097
Sequence in context: A035387 A011373 A117915 this_sequence A104726 A091224 A112182
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jul 26 2004
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