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, 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

Links

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Sela Fried, On an integer sequence related to Euler's formula for the Stirling numbers of the second kind, 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).

Cf. A007318, A048993, A004248, A343237.

Sequence in context: A275662 A110441 A111806 * A321163 A054458 A110168

Adjacent sequences: A372115 A372116 A372117 * A372119 A372120 A372121

Keyword

nonn,easy,tabl

Author

Werner Schulte, Apr 19 2024

Status

approved