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A178831
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Rectangular array T(n,k) = binomial(n+1,2)*(n^k - (n-1)^k) read by antidiagonals.
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1
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1, 1, 3, 1, 9, 6, 1, 21, 30, 10, 1, 45, 114, 70, 15, 1, 93, 390, 370, 135, 21, 1, 189, 1266, 1750, 915, 231, 28, 1, 381, 3990, 7810, 5535, 1911, 364, 36, 1, 765, 12354, 33670, 31515, 14091, 3556, 540, 45, 1, 1533, 37830, 141970, 172935, 97671, 30940, 6084, 765, 55
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OFFSET
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1,3
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COMMENTS
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T(n,k) is the sum of the elements in the image sets of all functions f:{1,2,...,k}->{1,2,...,n}.
Equivalently, the sum of the distinct entries in each length k sequence on {1,2,...,n}.
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LINKS
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FORMULA
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E.g.f. for row n: binomial(n+1,2)*exp((n-1)*x)*(exp(x) - 1).
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EXAMPLE
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Array begins
1, 1, 1, 1, 1, 1, ...
3, 9, 21, 45, 93, 189, ...
6, 30, 114, 390, 1266, 3990, ...
10, 70, 370, 1750, 7810, 33670, ...
15, 135, 915, 5535, 31515, 172935, ...
21, 231, 1911, 14091, 97671, 651651, ...
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MATHEMATICA
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Table[Range[7]! Rest[CoefficientList[Series[Binomial[n+1, 2] Exp[(n-1)x](Exp[x]-1), {x, 0, 7}], x]], {n, 1, 7}]//Grid
T[n_, k_]:= Binomial[n+2, 2]*((n+1)^k -n^k); Table[T[k, n-k], {n, 1, 10}, {k, 0, n-1}] (* G. C. Greubel, Jan 22 2019 *)
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PROG
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(PARI) {T(n, k) = binomial(n+2, 2)*((n+1)^k -(n)^k)};
for(n=1, 10, for(k=0, n-1, print1(T(k, n-k), ", "))) \\ G. C. Greubel, Jan 22 2019
(Magma) [[Binomial(k+2, 2)*((k+1)^(n-k) -k^(n-k)): k in [0..n-1]]: n in [1..10]]; // G. C. Greubel, Jan 22 2019
(Sage) [[binomial(k+2, 2)*((k+1)^(n-k) -k^(n-k)) for k in (0..n-1)] for n in (1..10)] # G. C. Greubel, Jan 22 2019
(GAP T:=Flat(List([1..10], n->List([0..n-1], k-> Binomial(k+2, 2)*( (k+1)^(n-k) -k^(n-k)) ))); # G. C. Greubel, Jan 22 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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