This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350149 #88 Aug 01 2022 08:12:16
%S A350149 1,1,1,4,2,2,27,9,6,6,256,64,32,24,24,3125,625,250,150,120,120,46656,
%T A350149 7776,2592,1296,864,720,720,823543,117649,33614,14406,8232,5880,5040,
%U A350149 5040,16777216,2097152,524288,196608,98304,61440,46080,40320,40320
%N A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.
%C A350149 T(n,k) are the denominators in a double summation power series for the definite integral of x^x. First expand x^x = exp(x*log(x)) = Sum_{n>=0} (x*log(x))^n/n!, then integrate each of the terms to get the double summation for F(x) = Integral_{t=0..x} t^t = Sum_{n>=1} (Sum_{k=0..n-1} (-1)^(n+k+1)*x^n*(log(x))^k/T(n,k)).
%C A350149 This is a definite integral, because lim {x->0} F(x) = 0.
%C A350149 The value of F(1) = 0.78343... = A083648 is known humorously as the Sophomore's Dream (see Borwein et al.).
%D A350149 Borwein, J., Bailey, D. and Girgensohn, R., Experimentation in Mathematics: Computational Paths to Discovery, A. K. Peters 2004.
%D A350149 William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton NJ 2005.
%H A350149 G. C. Greubel, <a href="/A350149/b350149.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A350149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SophomoresDream.html">Sophomore's dream</a>
%H A350149 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sophomore's_dream">Sophomore's dream</a>
%F A350149 T(n, 0) = A000312(n).
%F A350149 T(n, 1) = A000169(n).
%F A350149 T(n, 2) = A003308(n), n >= 2.
%F A350149 Sum_{k=0..n} T(n, k) = A112541(n).
%F A350149 T(n, n) = A000142(n).
%F A350149 T(n, n-1) = A000142(n), n >= 1.
%F A350149 T(n,k) = A061711(n) * (n+1) / A350297(n+1,k). - _Robert B Fowler_, Jan 11 2022
%e A350149 Triangle T(n,k) begins:
%e A350149 --------------------------------------------------------------------------
%e A350149 n/k 0 1 2 3 4 5 6 7 8
%e A350149 --------------------------------------------------------------------------
%e A350149 0 | 1,
%e A350149 1 | 1, 1,
%e A350149 2 | 4, 2, 2,
%e A350149 3 | 27, 9, 6, 6,
%e A350149 4 | 256, 64, 32, 24, 24,
%e A350149 5 | 3125, 625, 250, 150, 120, 120,
%e A350149 6 | 46656, 7776, 2592, 1296, 864, 720, 720,
%e A350149 7 | 823543, 117649, 33614, 14406, 8232, 5880, 5040, 5040,
%e A350149 8 | 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320.
%e A350149 ...
%p A350149 T := (n, k) -> n^(n - k)*k!:
%p A350149 seq(seq(T(n, k), k = 0..n), n = 0..9); # _Peter Luschny_, Jan 07 2022
%t A350149 T[n_, k_]:= n^(n-k)*k!; Table[T[n, k], {n, 0,12}, {k,0,n}]//Flatten (* _Amiram Eldar_, Dec 27 2021 *)
%o A350149 (Magma)
%o A350149 A350149:= func< n,k | n^(n-k)*Factorial(k) >;
%o A350149 [A350149(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 31 2022
%o A350149 (SageMath)
%o A350149 def A350149(n,k): return n^(n-k)*factorial(k)
%o A350149 flatten([[A350149(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 31 2022
%Y A350149 Cf. A000312 (first column), A000169 (2nd column), A003308 (3rd column excluding first term), A000142 (main diagonal), A000142 (2nd diagonal excluding first term), A112541 (row sums).
%Y A350149 Values of the integral: A083648, A073009.
%Y A350149 Cf. A061711, A350297.
%K A350149 easy,nonn,tabl
%O A350149 0,4
%A A350149 _Robert B Fowler_, Dec 27 2021