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Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...
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%I #28 Jan 03 2021 15:56:09

%S 1,1,1,2,2,1,6,5,3,1,24,16,10,4,1,120,65,38,17,5,1,720,326,168,78,26,

%T 6,1,5040,1957,872,393,142,37,7,1,40320,13700,5296,2208,824,236,50,8,

%U 1,362880,109601,37200,13977,5144,1569,366,65,9,1,3628800,986410,297856

%N Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function">Incomplete gamma function</a>.

%F a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).

%F a(n,k) = n*a(n-1,k) + k^n for n,k > 0.

%F E.g.f. (by columns) is e^(kx)/(1-x).

%F a(n,k) = the binomial transform by columns of a(n,k-1).

%F Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 2, 5, 10, 17, 26, 37, 50, ...

%e 6, 16, 38, 78, 142, 236, 366, ...

%e 24, 65, 168, 393, 824, 1569, 2760, ...

%e 120, 326, 872, 2208, 5144, 10970, 21576, ...

%e 720, 1957, 5296, 13977, 34960, 81445, 176112, ...

%t T[n_,k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* _Amiram Eldar_, Jun 27 2020 *)

%Y Cf. a(n, 0) = A000142(n); a(n, 1) = A000522(n); a(n, 2) = A010842(n); a(n, 3) = A053486(n); a(n, 4) = A053487(n); a(n, 5) = A080954(n); a(n, 6) = A108869(n); a(1, k) = A000027(k+1); a(2, k) = A002522(k+1); a(n, n) = A063170(n); a(n, n+1) = A001865(n+1); a(n, n+2) = A001863(n+2).

%Y Another version: A089258.

%Y A transposed version: A080955.

%Y Cf. A001113.

%K nonn,tabl

%O 0,4

%A _Ross La Haye_, Jan 22 2008

%E More terms from _Amiram Eldar_, Jun 27 2020