A292977 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x).

1, 1, 1, 1, 0, 2, 1, -1, 1, 6, 1, -2, 2, 2, 24, 1, -3, 5, -2, 9, 120, 1, -4, 10, -12, 8, 44, 720, 1, -5, 17, -34, 33, 8, 265, 5040, 1, -6, 26, -74, 120, -78, 112, 1854, 40320, 1, -7, 37, -138, 329, -424, 261, 656, 14833, 362880, 1, -8, 50, -232, 744, -1480, 1552, -360, 5504, 133496, 3628800

Offset: 0

Author

Ilya Gutkovskiy, Sep 27 2017

Keywords

• sign
• tabl

Comments

A(n,k) is the k-th inverse binomial transform of A000142 evaluated at n.

Can be considered as extension of the array A089258 to columns with negative indices via A089258(n,k) = A(n,-k) or, vice versa, A(n,k) = A089258(n,-k). - Max Alekseyev, Mar 06 2018

Examples

			Square array begins:
n=0:    1,   1,   1,    1,     1,      1,  ...
n=1:    1,   0,  -1,   -2,    -3,     -4,  ...
n=2:    2,   1,   2,    5,    10,     17,  ...
n=3:    6,   2,  -2,  -12,   -34,    -74,  ...
n=4:   24,   9,   8,   33,   120,    329,  ...
n=5:  120,  44,   8,  -78,  -424,  -1480,  ...
...
E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! +  (k^2 - 2*k + 2)*x^2/2! + (-k^3 + 3*k^2 - 6*k + 6) x^3/3! + (k^4 - 4*k^3 + 12*k^2 - 24*k + 24)*x^4/4! + ...
		

Links

• N. J. A. Sloane, Transforms

Crossrefs

Columns: A000142 (k=0), A000166 (k=1), A000023 (k=2), A010843 (k=3, with offset 0).

Main diagonal: A134095 (absolute values).

Cf. A080955, A089258.

Programs

Mathematica

Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
FullSimplify[Table[Function[k, Exp[-k] Gamma[n + 1, -k]][j - n], {j, 0, 10}, {n, 0, j}]] // Flatten

Formula

T(n, k) = n! * Sum_{j=0..n} (-k)^j/j!. - Max Alekseyev, Mar 06 2018

E.g.f. of column k: exp(-k*x)/(1 - x).