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A114202
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A Pascal-Jacobsthal triangle.
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4
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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;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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As a number triangle, with J(n) = A001045(n):
T(n, k) = Sum_{i=0..n-k} C(n-k, i)*C(k, i)*J(i);
T(n, k) = Sum_{i=0..n} C(n-k, n-i)*C(k, i-k)*J(i-k);
T(n, k) = Sum_{i=0..n} C(k, i)*C(n-k, n-i)*J(k-i) if k <= n, and 0 otherwise.
As a square array, with J(n) = A001045(n):
T(n, k) = Sum_{i=0..n} C(n, i)C(k, i)*J(i);
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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KEYWORD
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AUTHOR
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STATUS
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approved
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