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A372118
Square array A(n, k) = ((k+2)^(n+2) - 2 * (k+1)^(n+2) + k^(n+2))/2 for k, n >= 0 read by ascending antidiagonals.
1
1, 3, 1, 7, 6, 1, 15, 25, 9, 1, 31, 90, 55, 12, 1, 63, 301, 285, 97, 15, 1, 127, 966, 1351, 660, 151, 18, 1, 255, 3025, 6069, 4081, 1275, 217, 21, 1, 511, 9330, 26335, 23772, 9751, 2190, 295, 24, 1, 1023, 28501, 111645, 133057, 70035, 19981, 3465, 385, 27, 1
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
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
Rows: A000012 (n=0), A008585 (n=1), A227776 (n=2).
Columns: A000225 (k=0), A000392 (k=1), A016269 (k=2), A016753 (k=3), A016103 (k=4), A019757 (k=5), A020570 (k=6), A020782 (k=7).
Main diagonal: A281596(n+2).
Sequence in context: A275662 A110441 A111806 * A321163 A054458 A110168
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Apr 19 2024
STATUS
approved