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A299741 Array read by antidiagonals upwards: a(i,0) = 2, i >= 0; a(i,1) = i+2, i >= 0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), for i >= 0, j > 1.

%I A299741 #53 Apr 11 2021 09:53:06
%S A299741 2,2,2,2,3,2,2,4,7,2,2,5,14,18,2,2,6,23,52,47,2,2,7,34,110,194,123,2,
%T A299741 2,8,47,198,527,724,322,2,2,9,62,322,1154,2525,2702,843,2,2,10,79,488,
%U A299741 2207,6726,12098,10084,2207,2,2,11,98,702,3842,15127,39202,57965,37634,5778,2
%N A299741 Array read by antidiagonals upwards: a(i,0) = 2, i >= 0; a(i,1) = i+2, i >= 0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), for i >= 0, j > 1.
%C A299741 Note the similarity in form of the recursive steps in the array definition above and the polynomial definition under FORMULA.
%H A299741 William W. Collier, <a href="/A299741/a299741.txt">a(i,j) = f(i+2,j)</a>
%H A299741 William W. Collier, <a href="https://pok.acm.org/meetings/foils/Collier18.html">Experimental Mathematics on Wisteria Tables</a>, Talk to Poughkeepsie ACM Chapter.
%H A299741 OEIS Wiki, <a href="http://oeis.org/wiki/(1,2)-Pascal_triangle">The (1,2) Pascal Triangle</a>.
%F A299741 Let k be an integer, and let r1 and r2 be the roots of x + 1/x = k. Then f(k,n) = r1^n + r2^n is an integer, for integer n >= 0. Theorem: a(i,j) = f(i+2,j), for i,j >= 0. Proof: See the Collier link.
%F A299741 Define polynomials recursively by:
%F A299741     p[0](n) = 2, for n >= 0 ( [ and ] demark subscripts).
%F A299741     p[1](n) = n + 2, for n >= 0.
%F A299741     p[j](n) = p[j-1](n) * p[1](n) - p[j-2](n), for j > 1, n >= 0. The coefficients of these polynomials occur as the even numbered, upward diagonals in the OEIS Wiki link. Conjecture: a(i,j) = p[j](i), i,j >= 0.
%e A299741 i\j |0  1   2    3      4       5        6          7           8            9
%e A299741 ----+-------------------------------------------------------------------------
%e A299741    0|2  2   2    2      2       2        2          2           2            2
%e A299741    1|2  3   7   18     47     123      322        843        2207         5778
%e A299741    2|2  4  14   52    194     724     2702      10084       37634       140452
%e A299741    3|2  5  23  110    527    2525    12098      57965      277727      1330670
%e A299741    4|2  6  34  198   1154    6726    39202     228486     1331714      7761798
%e A299741    5|2  7  47  322   2207   15127   103682     710647     4870847     33385282
%e A299741    6|2  8  62  488   3842   30248   238142    1874888    14760962    116212808
%e A299741    7|2  9  79  702   6239   55449   492802    4379769    38925119    345946302
%e A299741    8|2 10  98  970   9602   95050   940898    9313930    92198402    912670090
%e A299741    9|2 11 119 1298  14159  154451  1684802   18378371   200477279   2186871698
%e A299741   10|2 12 142 1692  20162  240252  2862862   34114092   406506242   4843960812
%e A299741   11|2 13 167 2158  27887  360373  4656962   60180133   777684767  10049721838
%e A299741   12|2 14 194 2702  37634  524174  7300802  101687054  1416317954  19726764302
%e A299741   13|2 15 223 3330  49727  742575 11088898  165590895  2472774527  36926027010
%e A299741   14|2 16 254 4048  64514 1028176 16386302  261152656  4162056194  66331746448
%e A299741   15|2 17 287 4862  82367 1395377 23639042  400468337  6784322687 114933017342
%e A299741   16|2 18 322 5778 103682 1860498 33385282  599074578 10749957122 192900153618
%e A299741   17|2 19 359 6802 128879 2441899 46267202  876634939 16609796639 314709501202
%e A299741   18|2 20 398 7940 158402 3160100 63043598 1257711860 25091193602 500566160180
%e A299741   19|2 21 439 9198 192719 4037901 84603202 1772629341 37140612959 778180242798
%p A299741 A:= proc(i, j) option remember; `if`(min(i, j)=0, 2,
%p A299741       `if`(j=1, i+2, (i+2)*A(i, j-1)-A(i, j-2)))
%p A299741     end:
%p A299741 seq(seq(A(d-k, k), k=0..d), d=0..12);  # _Alois P. Heinz_, Mar 05 2019
%t A299741 a[_, 0] = a[0, _] = 2; a[i_, 1] := i + 2;
%t A299741 a[i_, j_] := a[i, j] =(i + 2) a[i, j - 1] - a[i, j - 2];
%t A299741 Table[a[i - j, j], {i, 0, 10}, {j, 0, i}] // Flatten (* _Jean-François Alcover_, Dec 07 2019 *)
%Y A299741 The array first appeared in A298675.
%Y A299741 Rows 1 through 29 of the array appear in these OEIS entries: A005248, A003500, A003501, A003499, A056854, A086903, A056918, A087799, A057076, A087800, A078363, A067902, A078365, A090727, A078367, A087215, A078369, A090728, A090729, A090730, A090731, A090732, A090733, A090247, A090248, A090249, A090251. Also entries occur for rows 45, 121, and 320:  A087265, A065705, A089775. Each of these entries asserts that a(i,j)=f(i+2,j) is true for that row.
%Y A299741 A few of the columns appear in the OEIS: A008865 (for column 2), A058794 and A007754 (for column 3), and A230586 (for column 5).
%Y A299741 Main diagonal gives A343261.
%K A299741 easy,nonn,tabl
%O A299741 0,1
%A A299741 _William W. Collier_, Feb 18 2018
%E A299741 Edited by _N. J. A. Sloane_, Apr 04 2018