A134558 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...
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, 6, 1, 5040, 1957, 872, 393, 142, 37, 7, 1, 40320, 13700, 5296, 2208, 824, 236, 50, 8, 1, 362880, 109601, 37200, 13977, 5144, 1569, 366, 65, 9, 1, 3628800, 986410, 297856
Offset: 0
Examples
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
2, 5, 10, 17, 26, 37, 50, ...
6, 16, 38, 78, 142, 236, 366, ...
24, 65, 168, 393, 824, 1569, 2760, ...
120, 326, 872, 2208, 5144, 10970, 21576, ...
720, 1957, 5296, 13977, 34960, 81445, 176112, ...
Links
- Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
- Wikipedia, Incomplete gamma function.
Crossrefs
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).
Another version: A089258.
A transposed version: A080955.
Cf. A001113.
Programs
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Mathematica
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 *)
Formula
a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).
a(n,k) = n*a(n-1,k) + k^n for n,k > 0.
E.g.f. (by columns) is e^(kx)/(1-x).
a(n,k) = the binomial transform by columns of a(n,k-1).
Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.
Extensions
More terms from Amiram Eldar, Jun 27 2020