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A178831 Rectangular array T(n,k) = binomial(n+1,2)*(n^k - (n-1)^k) read by antidiagonals.

%I A178831 #27 Oct 14 2024 02:33:05
%S A178831 1,1,3,1,9,6,1,21,30,10,1,45,114,70,15,1,93,390,370,135,21,1,189,1266,
%T A178831 1750,915,231,28,1,381,3990,7810,5535,1911,364,36,1,765,12354,33670,
%U A178831 31515,14091,3556,540,45,1,1533,37830,141970,172935,97671,30940,6084,765,55
%N A178831 Rectangular array T(n,k) = binomial(n+1,2)*(n^k - (n-1)^k) read by antidiagonals.
%C A178831 T(n,k) is the sum of the elements in the image sets of all functions f:{1,2,...,k}->{1,2,...,n}.
%C A178831 Equivalently, the sum of the distinct entries in each length k sequence on {1,2,...,n}.
%H A178831 G. C. Greubel, <a href="/A178831/b178831.txt">Antidiagonals n = 1..100, flattened</a>
%F A178831 E.g.f. for row n: binomial(n+1,2)*exp((n-1)*x)*(exp(x) - 1).
%e A178831 Array begins
%e A178831    1,   1,    1,     1,     1,      1, ...
%e A178831    3,   9,   21,    45,    93,    189, ...
%e A178831    6,  30,  114,   390,  1266,   3990, ...
%e A178831   10,  70,  370,  1750,  7810,  33670, ...
%e A178831   15, 135,  915,  5535, 31515, 172935, ...
%e A178831   21, 231, 1911, 14091, 97671, 651651, ...
%t A178831 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 A178831 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 *)
%o A178831 (PARI) {T(n, k) = binomial(n+2, 2)*((n+1)^k -(n)^k)};
%o A178831 for(n=1,10, for(k=0,n-1, print1(T(k,n-k), ", "))) \\ _G. C. Greubel_, Jan 22 2019
%o A178831 (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
%o A178831 (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
%o A178831 (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
%Y A178831 Cf. A068156 the case for n=2.
%K A178831 nonn,tabl,easy
%O A178831 1,3
%A A178831 _Geoffrey Critzer_, Dec 27 2010
%E A178831 Terms a(29) onward added by _G. C. Greubel_, Jan 22 2019