This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A136217 #9 Nov 06 2019 12:40:31
%S A136217 1,1,1,3,2,1,15,8,3,1,108,49,15,4,1,1036,414,108,24,5,1,12569,4529,
%T A136217 1036,198,34,6,1,185704,61369,12569,2116,306,46,7,1,3247546,996815,
%U A136217 185704,28052,3493,453,59,8,1,65762269,18931547,3247546,446560,48800,5555,622,74,9,1
%N A136217 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {floor(m*(m+7)/6), m>=0} and then taking partial sums, starting with all 1's in row 0.
%C A136217 A variant of the triple factorial array A136212. Compare to triangle array A136218, which is generated by a complementary process.
%F A136217 Let triangular matrix P = A136220, then: column 0 (A136221) = column 0 of P; column 1 (A136226) = column 0 of P^2; column 3 (A136229) = column 0 of P^4.
%e A136217 Square array begins:
%e A136217 (1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),...;
%e A136217 (1),(2),3,(4),5,(6),7,(8),9,10,(11),12,13,(14),15,16,(17),18,19,20,(21),..;
%e A136217 (3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),216,242,(269),..;
%e A136217 (15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),2062,2485,(2943),..;
%e A136217 (108),(414),1036,(2116),3493,(5555),8040,(11477),15483,20748,(26748),33528,..;
%e A136217 (1036),(4529),12569,(28052),48800,(82328),124335,(186261),260856,364551,..;
%e A136217 (12569),(61369),185704,(446560),811111,(1438447),2250731,(3513569),5078154,..;
%e A136217 (185704),(996815),3247546,(8325700),15684001,(29039188),46830722,...;
%e A136217 (3247546),(18931547),65762269,(178284892),346583419,...;
%e A136217 (65762269),(412345688),1515642725,(4317391240),...; ...
%e A136217 where terms in parenthesis are at positions {floor(m*(m+7)/6), m>=0} and are removed before taking partial sums to obtain the next row.
%e A136217 To generate the array, start with all 1's in row 0; from then on, obtain row n+1 from row n by first removing terms in row n at positions {floor(m*(m+7)/6), m>=0} and then taking partial sums.
%e A136217 For example, to generate row 2 from row 1:
%e A136217 [(1),(2),3,(4),5,(6),7,(8),9,10,(11),12,13,(14),15,16,(17),18,...],
%e A136217 remove terms at positions [0,1,3,5,7,10,13,16,20,...] to get:
%e A136217 [3, 5, 7, 9,10, 12,13, 15,16, 18,19,20, 22,23,24, 26,27,28,...]
%e A136217 then take partial sums to obtain row 2:
%e A136217 [3,8,15,24,34,46,59,74,90,108,127,147,169,192,216,242,269,...].
%e A136217 Continuing in this way will generate all the rows of this array.
%e A136217 Amazingly, column 0 of this array = column 0 of triangle P=A136220:
%e A136217 1;
%e A136217 1, 1;
%e A136217 3, 2, 1;
%e A136217 15, 10, 3, 1;
%e A136217 108, 75, 21, 4, 1;
%e A136217 1036, 753, 208, 36, 5, 1;
%e A136217 12569, 9534, 2637, 442, 55, 6, 1;
%e A136217 185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
%e A136217 where column k of P^3 = column 0 of P^(3k+3) such that column 0 of P^3 = column 0 of P shift one place left.
%t A136217 nmax = 9;
%t A136217 row[0] = Table[1, {nmax^2}];
%t A136217 row[n_] := row[n] = Accumulate[Delete[row[n-1], Table[{Floor[m((m+7)/6)+1] }, {m, 0, (1/2)(-7 + Sqrt[1 + 24 Length[row[n-1]]]) // Floor}]]];
%t A136217 R = row /@ Range[0, nmax];
%t A136217 T[n_, k_] := R[[n+1, k+1]];
%t A136217 Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 06 2019 *)
%o A136217 (PARI) {T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+7))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}
%Y A136217 Cf. columns: A136221, A136226, A136229; related tables: A136220 (P), A136226 (P^2), A136232 (P^4).
%K A136217 nice,nonn,tabl
%O A136217 0,4
%A A136217 _Paul D. Hanna_, Dec 23 2007