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A116924
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Triangle T(n,k)= B(k,n)-B(k-1,n) where B(n,m) = sum_{i=0..n} binomial(m,i) (3*i+1).
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1
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1, 1, 4, 1, 8, 7, 1, 12, 21, 10, 1, 16, 42, 40, 13, 1, 20, 70, 100, 65, 16, 1, 24, 105, 200, 195, 96, 19, 1, 28, 147, 350, 455, 336, 133, 22, 1, 32, 196, 560, 910, 896, 532, 176, 25, 1, 36, 252, 840, 1638, 2016, 1596, 792, 225, 28, 1, 40, 315, 1200
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The auxiliary array B(n,m) contains the binomial transform of the finite sequence of the first n+1 terms
of 1, 4, 7, .., = A016777(.) in row n.
First few rows of the array B(n,m) are from row n=0 on:
1, 1, 1, 1, 1,... (binomial transform of 1,0,0,0....)
1, 5, 9, 13, 17,... (binomial transform of 1,4,0,0....)
1, 5, 16, 34, 59,... (binomial transform of 1,4,7,0....)
1, 5, 16, 44, 99,... (binomial transform of 1,4,7,10,0,0....)
The n-th row of triangle T contains the first differences of the n-th column of B.
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FORMULA
| sum_{k=0..n} T(n,k) = A053220(n+1).
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EXAMPLE
| First few rows of the triangle T(n,k) are:
1;
1, 4;
1, 8, 7;
1, 12, 21, 10;
1, 16, 42, 40, 13;
1, 20, 70, 100, 65, 16;
1, 24, 105, 200, 195, 96, 19;
...
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MAPLE
| B := proc(n, m) add(binomial(m, i)*(3*i+1), i=0..n) ; end proc:
A116924 := proc(n, k) if k = 0 then 1; else B(k, n)-B(k-1, n) ; end if; end proc:
seq(seq(A116924(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Mar 27 2010
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CROSSREFS
| Cf. A016777, A053220.
Sequence in context: A143820 A103553 A067439 * A128414 A192014 A019699
Adjacent sequences: A116921 A116922 A116923 * A116925 A116926 A116927
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KEYWORD
| nonn,tabl,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 26 2006
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EXTENSIONS
| Definition replaced by formula, corrected from 5th row on - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 27 2010
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