A369435 Square array A(n, k) = n! * [t^n] (exp(t)/(1+k-k*exp(t))) for n >= 0 and k >= 0, read by antidiagonals upwards.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 26, 15, 4, 1, 1, 150, 111, 28, 5, 1, 1, 1082, 1095, 292, 45, 6, 1, 1, 9366, 13503, 4060, 605, 66, 7, 1, 1, 94586, 199815, 70564, 10845, 1086, 91, 8, 1, 1, 1091670, 3449631, 1471708, 243005, 23826, 1771, 120, 9, 1, 1, 14174522, 68062695, 35810212, 6534045, 653406, 45955, 2696, 153, 10, 1

Offset: 0

Author

Werner Schulte, Jan 23 2024

Keywords

• nonn
• easy
• tabl

Comments

The following formulae are conjectures:

(1) det(A(0..n, k..k+n)) = (Product_{i=1..n} i!)^2 for k >= 0 and n >= 0.

(2) A(n, k) = 1 + k * (Sum_{i=0..n-1} binomial(n, i) * A(i, k)) for k >= 0 and

n > 0 with initial values A(0, k) = 1 for k >= 0.

(3) A(n, k) = (k+1)^n + k * (Sum_{i=0..n-2} binomial(n, i) * A(i, k) *

((k+1)^(n-i) - (k+1) * k^(n-1-i))) for k >= 0 and n > 1 with initial values

A(n, k) = (k+1)^n for k >= 0 and n < 2.

(4) Let B(n, k) = (k!) * (Sum_{i=k..n} (i!) * S2(i, k) * S2(n+1, i+1)) for 0 <=

k <= n where S2(i, j) = A048993(i, j). Then holds:

(a) B(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A(n, i) for 0 <= k

<= n;

(b) E.g.f. of row n >= 0: exp(x) * (Sum_{k=0..n} B(n, k) * x^k / (k!)).

Examples

			Array A(n, k) starts:
n\k : 0        1        2        3       4        5        6         7         8
================================================================================
 0  : 1        1        1        1       1        1        1         1         1
 1  : 1        2        3        4       5        6        7         8         9
 2  : 1        6       15       28      45       66       91       120       153
 3  : 1       26      111      292     605     1086     1771      2696      3897
 4  : 1      150     1095     4060   10845    23826    45955     80760    132345
 5  : 1     1082    13503    70564  243005   653406  1490587   3024008   5618169
 6  : 1     9366   199815  1471708 6534045 21502866 58018051 135878520 286195833
 7  : 1    94586  3449631 35810212
 8  : 1  1091670 68062695
 9  : 1 14174522
.
Triangle T(n, k) starts:
[0] 1;
[1] 1,       1;
[2] 1,       2,       1;
[3] 1,       6,       3,       1;
[4] 1,      26,      15,       4,      1;
[5] 1,     150,     111,      28,      5,     1;
[6] 1,    1082,    1095,     292,     45,     6,   1;
[7] 1,    9366,   13503,    4060,    605,    66,   7,    1;
[8] 1,   94586,  199815,   70564,  10845,  1086,   91,   8, 1;
[9] 1, 1091670, 3449631, 1471708, 243005, 23826, 1771, 120, 9, 1;
		

Crossrefs

Cf. A000012 (col 0 and row 0), A000629 (col 1), A201339 (col 2), A201354 (col 3), A201365 (col 4), A000027 (row 1), A000384 (row 2), A163626, A028246.

Cf. A372312.

Programs

Maple

egf := exp(t) / (1 + x*(1 - exp(t))): sert := series(egf, t, 12):
col := k -> local j; seq(subs(x=k, j!*coeff(sert, t, j)), j=0..9):
T := (n, k) -> col(k)[n - k + 1]:  # Triangle
for n from 0 to 9 do seq(T(n, k), k=0..n) od;  # Peter Luschny, Jan 24 2024
with(combinat): # WP Worpitzky polynomials, WC coefficients of WP.
WC := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j=0..n):
WP := n -> local j; add(WC(n, j) * x^j, j=0..n):
A369435row := (n, k) -> subs(x = k, WP(n)):
seq(lprint(seq(A369435row(n, k), k = 0..7)), n = 0..7);
# Peter Luschny, Apr 26 2024

PARI

{A(n, k) = n! * polcoeff(exp(x+x*O(x^n)) / (1+k-k*exp(x+x*O(x^n))), n)}

Formula

A(n, k) = Sum_{i=0..n} A163626(n, i) * (-k)^i for n >= 0 and k >= 0.

A(n, k) = Sum_{i=0..n} A028246(n+1, i+1) * k^i for n >= 0 and k >= 0.

E.g.f. of column k >= 0: exp(t) / (1 + k - k * exp(t)).

A(n, n) = Sum_{i=0..n} A163626(n, i) * (-n)^i = Sum_{i=0..n} A028246(n+1, i+1) * n^i for n >= 0.

Conjecture: A(n, n) = (n + 1) * A321189(n) for n >= 0. [This is true. - Peter Luschny, Apr 26 2024]

A(n, n) = A372312(n). - Peter Luschny, Apr 26 2024