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A113582 A symmetrical triangle of coefficients: t(n,m)=(n - m)*(n - m + 1)*m*(m + 1)/4 + 1. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..66.

FORMULA

t(n,m)=(n - m)*(n - m + 1)*m*(m + 1)/4 + 1.

EXAMPLE

{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}

MATHEMATICA

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[%]

CROSSREFS

Sequence in context: A161126 A128562 A034368 * A118245 A104382 A086629

Adjacent sequences:  A113579 A113580 A113581 * A113583 A113584 A113585

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Aug 25 2008

STATUS

approved

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Last modified November 23 04:44 EST 2014. Contains 249839 sequences.