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A099239
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Square array read by anti-diagonals associated with sections of 1/(1-x-x^k).
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4
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1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 21, 13, 5, 1, 1, 32, 55, 41, 19, 6, 1, 1, 64, 144, 129, 69, 26, 7, 1, 1, 128, 377, 406, 250, 106, 34, 8, 1, 1, 256, 987, 1278, 907, 431, 153, 43, 9, 1, 1, 512, 2584, 4023, 3292, 1757, 686, 211, 53, 10, 1, 1, 1024, 6765, 12664
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Rows include A099242, A099253. Columns include A034856. Main diagonal is A099240. Sums of antidiagonals are A099241.
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FORMULA
| Square array T(n, k)=sum{j=0..n, binomial(k*n-(k-1)(j-1), j)}, n, k>=0. Also, T(n, k)=sum{j=0..n, binomial(k+(n-1)(j+1), n(j+1)-1}, n>0. As a number triangle read by row, this is T(n, k)=sum{j=0..n-k, binomial(k(n-k)-(k-1)(j-1)}. Rows of the square array are generated by 1/((1-x)^k-x). Rows satisfy a(n)=a(n-1)-sum{k=1..n, (-1)^k^C(n, k)a(n-k)}.
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EXAMPLE
| Rows begin
1,1,1,1,1,1,1,...
1,2,4,8,16,32,... 1-section of 1/(1-x-x) A000079
1,3,8,21,55,..... bisection of 1/(1-x-x^2) A001906
1,4,13,41,129,... trisection of 1/(1-x-x^3) A052529 (essentially)
1,5,19,69,250,... quadrisection of 1/(1-x-x^4) A055991
1,6,26,106,431,.. quintisection of 1/(1-x-x^5) A079675 (essentially)
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CROSSREFS
| Sequence in context: A138155 A055587 A137743 * A167630 A009998 A113993
Adjacent sequences: A099236 A099237 A099238 * A099240 A099241 A099242
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Oct 08 2004
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