OFFSET
0,2
COMMENTS
Depending on some fixed integer m >= 0 we define a family of square arrays A(m; n, k) = (Sum_{i=0..m} (-1)^i * binomial(m, i) * (k + m - i)^(n+m)) / m! for k, n >= 0. Special cases are: A004248 (m=0), A343237 (m=1) and this array (m=2). The A(m; n, k) satisfy: A(m; n, k) = (k+m) * A(m; n-1, k) + A(m-1; n, k) with initial values A(0; n, k) = k^n and A(m; 0, k) = 1.
Further properties are conjectures:
(1) O.g.f. of column k is Prod_{i=k..k+m} 1 / (1 - i * t);
(2) E.g.f. of row n is exp(x) * (Sum_{k=0..n} binomial(k+m, m) * A048993(n+m, k+m) * x^k);
(3) The LU decompositions of these arrays are given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(m; n, k) = A048993(n+m, k+m) * (k+m)! / m!, i.e., A(m; n, k) = Sum_{i=0..k} L(m; n, i) * binomial(k, i).
The three conjectures are true, see links. - Sela Fried, Jul 07 2024
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
FORMULA
A(n, k) = (k+2) * A(n-1, k) + (k+1)^(n+1) - k^(n+1) for n > 0.
Conjectures:
(1) O.g.f. of column k is Prod_{i=k..k+2} 1 / (1 - i * t);
(2) E.g.f. of row n is exp(x) * (Sum_{k=0..n} binomial(k+2, 2) * A048993(n+2, k+2) * x^k);
(3) The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(n, k) = A048993(n+2, k+2) * (k+2)! / 2!, i.e., A(n, k) = Sum_{i=0..k} L(n, i) * binomial(k, i).
The three conjectures are true. See comments. - Sela Fried, Jul 09 2024
EXAMPLE
Square array A(n, k) starts:
n\k : 0 1 2 3 4 5 6 7
=======================================================================
0 : 1 1 1 1 1 1 1 1
1 : 3 6 9 12 15 18 21 24
2 : 7 25 55 97 151 217 295 385
3 : 15 90 285 660 1275 2190 3465 5160
4 : 31 301 1351 4081 9751 19981 36751 62401
5 : 63 966 6069 23772 70035 170898 365001 706104
6 : 127 3025 26335 133057 481951 1398097 3463615 7628545
7 : 255 9330 111645 724260 3216795 11075670 31794105 79669320
etc.
MATHEMATICA
A372118[n_, k_] := ((k+2)^(n+2) - 2*(k+1)^(n+2) + k^(n+2))/2;
Table[A372118[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jul 10 2024 *)
PROG
(PARI) A(n, k) = ((k+2)^(n+2) - 2 * (k+1)^(n+2) + k^(n+2))/2
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Apr 19 2024
STATUS
approved