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Search: id:A114202
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| A114202 |
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A Pascal-Jacobsthal triangle. |
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+0
5
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| 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 16, 16, 5, 1, 1, 6, 27, 42, 27, 6, 1, 1, 7, 41, 87, 87, 41, 7, 1, 1, 8, 58, 156, 216, 156, 58, 8, 1, 1, 9, 78, 254, 456, 456, 254, 78, 9, 1, 1, 10, 101, 386, 860, 1122, 860, 386, 101, 10, 1
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row sums are A114203. T(2n,n) is A114204. Inverse has row sums 0^n.
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REFERENCES
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Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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FORMULA
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As a number triangle, T(n, k)=sum{i=0..n-k, C(n-k, i)C(k, i)J(i)} where J(n)=A001045(n); As a number triangle, T(n, k)=sum{i=0..n, C(n-k, n-i)C(k, i-k)J(i-k)}; As a number triangle, T(n, k)=if(k<=n, sum{i=0..n, C(k, i)C(n-k, n-i)J(k-i)}, 0); As a square array, T(n, k)=sum{i=0..n, C(n, i)C(k, i)J(i)}; As a square array, T(n, k)=sum{i=0..n+k, C(n, n+k-i)C(k, i-k)J(i-k)}; Column k has g.f. sum{i=0..k, C(k, i)J(i+1)(x/(1-x))^i}x^k/(1-x).
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EXAMPLE
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Triangle begins
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 8, 4, 1;
1, 5, 16, 16, 5, 1;
1, 6, 27, 42, 27, 6, 1;
1, 7, 41, 87, 87, 41, 7, 1;
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CROSSREFS
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Sequence in context: A135597 A073134 A026692 this_sequence A125806 A156354 A099597
Adjacent sequences: A114199 A114200 A114201 this_sequence A114203 A114204 A114205
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Nov 16 2005
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