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 A233531 #13 Dec 11 2013 23:29:17
%S A233531 1,1,-2,6,-18,44,-56,-300,2024,-22022,-130456,-4241064,-103538532,
%T A233531 -2893308780,-88314189664,-2924814872208,-104538530634844,
%U A233531 -4010605941377292,-164409679858874856,-7172735079437282200,-331847552362286195156,-16229743737669369558956,-836695536495554388520400
%N A233531 G.f. A(x) such that triangle A233530, which transforms diagonals in the table of successive iterations of A(x), consists of all zeros after row 1.
%e A233531 G.f.: A(x) = x + x^2 - 2*x^3 + 6*x^4 - 18*x^5 + 44*x^6 - 56*x^7 - 300*x^8 + 2024*x^9 - 22022*x^10 - 130456*x^11 - 4241064*x^12 - 103538532*x^13 - 2893308780*x^14 - 88314189664*x^15 - 2924814872208*x^16 +...
%e A233531 If we form a table of coefficients in the iterations of A(x) like so:
%e A233531 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e A233531 [1, 1, -2, 6, -18, 44, -56, -300, 2024, -22022, ...];
%e A233531 [1, 2, -2, 3, 2, -48, 228, -734, -1298, -14630, ...];
%e A233531 [1, 3, 0, -3, 18, -54, -24, 625, -6324, -46064, ...];
%e A233531 [1, 4, 4, -6, 12, 26, -332, 244, -2078, -108754, ...];
%e A233531 [1, 5, 10, 0, -10, 90, -192, -2044, -3190, -137176, ...];
%e A233531 [1, 6, 18, 21, -18, 54, 312, -3178, -22032, -203692, ...];
%e A233531 [1, 7, 28, 63, 42, -28, 616, -931, -46722, -457746, ...];
%e A233531 [1, 8, 40, 132, 248, 156, 504, 3144, -51348, -913356, ...];
%e A233531 [1, 9, 54, 234, 702, 1296, 1656, 6924, -24444, -1366530, ...];
%e A233531 [1, 10, 70, 375, 1530, 4580, 9916, 22122, 38570, -1538042, ...];
%e A233531 [1, 11, 88, 561, 2882, 11814, 38280, 104929, 273592, -987932, ...];
%e A233531 [1, 12, 108, 798, 4932, 25542, 110604, 407932, 1351614, 2563858, ...]; ...
%e A233531 then the triangle A233530, that transforms one diagonal in the above table into another, consists of all zeros in column 0 after row 1:
%e A233531 1;
%e A233531 1, 1;
%e A233531 0, 2, 1;
%e A233531 0, 3, 3, 1;
%e A233531 0, 8, 9, 4, 1;
%e A233531 0, 38, 40, 18, 5, 1;
%e A233531 0, 268, 264, 112, 30, 6, 1;
%e A233531 0, 2578, 2379, 953, 240, 45, 7, 1;
%e A233531 0, 31672, 27568, 10500, 2505, 440, 63, 8, 1;
%e A233531 0, 475120, 392895, 143308, 32686, 5445, 728, 84, 9, 1;
%e A233531 0, 8427696, 6663624, 2342284, 514660, 82176, 10423, 1120, 108, 10, 1;
%e A233531 0, 172607454, 131211423, 44677494, 9514570, 1467837, 178689, 18214, 1632, 135, 11, 1; ...
%e A233531 Illustrate how T=A233530 transforms one diagonal in the above table into another:
%e A233531 T*[1, 1, -2, -3, 12, 90, 312, -931, -51348, -1366530, ...]
%e A233531 = [1, 2, 0, -6, -10, 54, 616, 3144, -24444, -1538042, ...];
%e A233531 T*[1, 2, 0, -6, -10, 54, 616, 3144, -24444, -1538042, ...]
%e A233531 = [1, 3, 4, 0, -18,-28, 504, 6924, 38570, -987932, ...];
%e A233531 T*[1, 3, 4, 0, -18,-28, 504, 6924, 38570, -987932, ...]
%e A233531 = [1, 4,10, 21, 42,156,1656,22122, 273592, 2563858, ...].
%o A233531 (PARI) /* Given A = g.f. A(x), Calculate Triangle A233530: */
%o A233531 {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x;
%o A233531 for(i=1, r+c-2, F=subst(F, x, A +x*O(x^(m+2)))); polcoeff(F, c));
%o A233531 N=matrix(m+1, m+1, r, c, M[r, c]);
%o A233531 P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%o A233531 /* Calculates A = g.f. A(x) and then Prints ROWS of Triangle: */
%o A233531 {ROWS=20;V=[1,1];print("");print1("This Sequence: [1, 1, ");
%o A233531 for(i=2,ROWS,V=concat(V,0);A=x*truncate(Ser(V));
%o A233531 for(n=0,#V-1,if(n==#V-1,V[#V]=-T(n,0));for(k=0,n, T(n,k)));print1(V[#V]", "););
%o A233531 print1("...]");print("");print("");print("Triangle A233530 begins:");
%o A233531 for(n=0,#V-2,for(k=0,n,print1(T(n,k),", "));print(""))}
%Y A233531 Cf. A233530.
%K A233531 sign
%O A233531 1,3
%A A233531 _Paul D. Hanna_, Dec 11 2013