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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.

%I A372118 #32 Jul 10 2024 10:00:45
%S A372118 1,3,1,7,6,1,15,25,9,1,31,90,55,12,1,63,301,285,97,15,1,127,966,1351,
%T A372118 660,151,18,1,255,3025,6069,4081,1275,217,21,1,511,9330,26335,23772,
%U A372118 9751,2190,295,24,1,1023,28501,111645,133057,70035,19981,3465,385,27,1
%N 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.
%C A372118 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.
%C A372118 Further properties are conjectures:
%C A372118   (1) O.g.f. of column k is Prod_{i=k..k+m} 1 / (1 - i * t);
%C A372118   (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);
%C A372118   (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).
%C A372118 The three conjectures are true, see links. - _Sela Fried_, Jul 07 2024
%H A372118 Paolo Xausa, <a href="/A372118/b372118.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).
%H A372118 Sela Fried, <a href="/A372118/a372118_3.pdf">On an integer sequence related to Euler's formula for the Stirling numbers of the second kind</a>, 2024.
%F A372118 A(n, k) = (k+2) * A(n-1, k) + (k+1)^(n+1) - k^(n+1) for n > 0.
%F A372118 Conjectures:
%F A372118   (1) O.g.f. of column k is Prod_{i=k..k+2} 1 / (1 - i * t);
%F A372118   (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);
%F A372118   (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).
%F A372118 The three conjectures are true. See comments. - _Sela Fried_, Jul 09 2024
%e A372118 Square array A(n, k) starts:
%e A372118 n\k :    0     1       2       3        4         5         6         7
%e A372118 =======================================================================
%e A372118   0 :    1     1       1       1        1         1         1         1
%e A372118   1 :    3     6       9      12       15        18        21        24
%e A372118   2 :    7    25      55      97      151       217       295       385
%e A372118   3 :   15    90     285     660     1275      2190      3465      5160
%e A372118   4 :   31   301    1351    4081     9751     19981     36751     62401
%e A372118   5 :   63   966    6069   23772    70035    170898    365001    706104
%e A372118   6 :  127  3025   26335  133057   481951   1398097   3463615   7628545
%e A372118   7 :  255  9330  111645  724260  3216795  11075670  31794105  79669320
%e A372118   etc.
%t A372118 A372118[n_, k_] := ((k+2)^(n+2) - 2*(k+1)^(n+2) + k^(n+2))/2;
%t A372118 Table[A372118[n-k, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jul 10 2024 *)
%o A372118 (PARI) A(n, k) = ((k+2)^(n+2) - 2 * (k+1)^(n+2) + k^(n+2))/2
%Y A372118 Rows: A000012 (n=0), A008585 (n=1), A227776 (n=2).
%Y A372118 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).
%Y A372118 Main diagonal: A281596(n+2).
%Y A372118 Cf. A007318, A048993, A004248, A343237.
%K A372118 nonn,easy,tabl
%O A372118 0,2
%A A372118 _Werner Schulte_, Apr 19 2024