OFFSET
0,5
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!)).
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)).
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
EXAMPLE
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;
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
PROG
(PARI) {A(n, k) = n! * polcoeff(exp(x+x*O(x^n)) / (1+k-k*exp(x+x*O(x^n))), n)}
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Jan 23 2024
STATUS
approved