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A113582
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A symmetrical triangle of coefficients: t(n,m)=(n - m)*(n - m + 1)*m*(m + 1)/4 + 1.
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3
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1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 10, 7, 1, 1, 11, 19, 19, 11, 1, 1, 16, 31, 37, 31, 16, 1, 1, 22, 46, 61, 61, 46, 22, 1, 1, 29, 64, 91, 101, 91, 64, 29, 1, 1, 37, 85, 127, 151, 151, 127, 85, 37, 1, 1, 46, 109, 169, 211, 226, 211, 169, 109, 46, 1
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OFFSET
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1,5
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COMMENTS
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Row sums are:
{1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298}.
Contribution from Paul Barry, Jan 07 2009: (Start)
This triangle follows a general construction method as follows: Let a(n) be an integer sequence
with a(0)=1, a(1)=1. Then T(n,k,r):=[k<=n](1+r*a(k)*a(n-k)) defines a symmetrical triangle.
Row sums are n+1+r*sum{k=0..n, a(k)*a(n-k)} and central coefficients are 1+r*a(n)^2.
Here a(n)=C(n+1,2) and r=1.
Row sums are A154322 and central coefficients are A154323. (End)
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LINKS
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Table of n, a(n) for n=1..66.
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FORMULA
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t(n,m)=(n - m)*(n - m + 1)*m*(m + 1)/4 + 1.
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EXAMPLE
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{1},
{1, 1},
{1, 2, 1},
{1, 4, 4, 1},
{1, 7, 10, 7, 1},
{1, 11, 19, 19, 11, 1},
{1, 16, 31, 37, 31, 16, 1},
{1, 22, 46, 61, 61, 46, 22, 1},
{1, 29, 64, 91, 101, 91, 64, 29, 1},
{1, 37, 85, 127, 151, 151, 127, 85, 37, 1},
{1, 46, 109, 169, 211, 226, 211, 169, 109, 46, 1}
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MATHEMATICA
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Clear[t, n, m] t[n_, m_] = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]
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CROSSREFS
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Sequence in context: A161126 A128562 A034368 * A118245 A104382 A086629
Adjacent sequences: A113579 A113580 A113581 * A113583 A113584 A113585
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KEYWORD
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nonn,tabl
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AUTHOR
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Roger L. Bagula, Aug 25 2008
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STATUS
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approved
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