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A072024
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Table by antidiagonals of T(n,k) = ((n+1)^k - (-n)^k)/(2*n+1).
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5
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0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 7, 1, 1, 0, 1, 11, 13, 13, 1, 1, 0, 1, 21, 55, 25, 21, 1, 1, 0, 1, 43, 133, 181, 41, 31, 1, 1, 0, 1, 85, 463, 481, 461, 61, 43, 1, 1, 0, 1, 171, 1261, 2653, 1281, 991, 85, 57, 1, 1, 0, 1, 341, 4039, 8425, 10501, 2821, 1891, 113, 73, 1, 1, 0
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OFFSET
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0,12
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COMMENTS
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Rows of the array have g.f. x/((1+k*x)*(1-(k+1)*x)). - Philippe Deléham, Nov 24 2013
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LINKS
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FORMULA
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EXAMPLE
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Rows start:
0 1 1 1 1 1 1 1 1 1 ...
0 1 1 3 5 11 21 43 85 171 ...
0 1 1 7 13 55 133 463 1261 4039 ...
0 1 1 13 25 181 481 2653 8425 40261 ...
0 1 1 21 41 461 1281 10501 36121 246141 ...
0 1 1 31 61 991 2821 32551 117181 1093711 ...
0 1 1 43 85 1891 5461 84883 314245 3879331 ...
0 1 1 57 113 3305 9633 194713 734161 11638089 ...
...
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MAPLE
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seq(seq( ((k+1)^(n-k) - (-k)^(n-k))/(2*k+1), k=0..n), n=0..12); # G. C. Greubel, Jan 27 2020
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MATHEMATICA
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T[n_, k_]:= ((n + 1)^k - (-n)^k)/(2n + 1); Flatten[Join[{0}, Table[T[k, n- k], {n, 1, 15}, {k, 0, n}]]] (* Indranil Ghosh, Mar 27 2017 *)
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PROG
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(PARI)
for(n=0, 10, for(k=0, 9, print1(((n+1)^k-(-n)^k)/(2*n+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017
(Magma) [((k+1)^(n-k) - (-k)^(n-k))/(2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 27 2020
(Sage)
def T(n, k): return ((n+1)^k - (-n)^k)/(2*n+1)
[[T(k, n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020
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CROSSREFS
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Rows include A057427, A001045, A015441, A053404, A053428, A053430, A065874, etc. Columns include A000004, A000012, A000012, A002061, A001844, A072025, etc.
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KEYWORD
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AUTHOR
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STATUS
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approved
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