A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.

1, 1, 1, 4, 2, 2, 27, 9, 6, 6, 256, 64, 32, 24, 24, 3125, 625, 250, 150, 120, 120, 46656, 7776, 2592, 1296, 864, 720, 720, 823543, 117649, 33614, 14406, 8232, 5880, 5040, 5040, 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320

Offset

0,4

Comments

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)).

This is a definite integral, because lim {x->0} F(x) = 0.

The value of F(1) = 0.78343... = A083648 is known humorously as the Sophomore's Dream (see Borwein et al.).

References

Borwein, J., Bailey, D. and Girgensohn, R., Experimentation in Mathematics: Computational Paths to Discovery, A. K. Peters 2004.

William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton NJ 2005.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Eric Weisstein's World of Mathematics, Sophomore's dream

Wikipedia, Sophomore's dream

Formula

T(n, 0) = A000312(n).

T(n, 1) = A000169(n).

T(n, 2) = A003308(n), n >= 2.

Sum_{k=0..n} T(n, k) = A112541(n).

T(n, n) = A000142(n).

T(n, n-1) = A000142(n), n >= 1.

T(n,k) = A061711(n) * (n+1) / A350297(n+1,k). - Robert B Fowler, Jan 11 2022

Example

Triangle T(n,k) begins:
--------------------------------------------------------------------------
n/k         0        1       2       3      4      5      6      7      8
--------------------------------------------------------------------------
0  |        1,
1  |        1,       1,
2  |        4,       2,      2,
3  |       27,       9,      6,      6,
4  |      256,      64,     32,     24,    24,
5  |     3125,     625,    250,    150,   120,   120,
6  |    46656,    7776,   2592,   1296,   864,   720,   720,
7  |   823543,  117649,  33614,  14406,  8232,  5880,  5040,  5040,
8  | 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320.
...

Maple

T := (n, k) -> n^(n - k)*k!:
seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jan 07 2022

Mathematica

T[n_, k_]:= n^(n-k)*k!; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Amiram Eldar, Dec 27 2021 *)

Prog

(Magma)
A350149:= func< n, k | n^(n-k)*Factorial(k) >;
[A350149(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2022
(SageMath)
def A350149(n, k): return n^(n-k)*factorial(k)
flatten([[A350149(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 31 2022

Crossrefs

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).

Values of the integral: A083648, A073009.

Cf. A061711, A350297.

Sequence in context: A075418 A199221 A096870 * A350609 A261253 A328334

Adjacent sequences: A350146 A350147 A350148 * A350150 A350151 A350152

Keyword

easy,nonn,tabl

Author

Robert B Fowler, Dec 27 2021

Status

approved