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.
Triangle begins: 1; 1, 1; 1, 2, 2, 1; 1, 3, 6, 9, 10, 8, 4, 1; 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1; 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...; 1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; 1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...; 1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...; 1, 9, 72, 540, 3864, 26628, 177744, 1153740, 7303164, 45179508,...; 1, 10, 90, 765, 6240, 49260, 378312, 2836548, 20817588,...; ... Multiplying the g.f. of column k by (1-x)^k, k>=1, with leading zeros, yields the g.f. of row k in the triangle A122890: 1; 0, 1; 0, 0, 2; 0, 0, 1, 5; 0, 0, 0, 10, 14; 0, 0, 0, 8, 70, 42; 0, 0, 0, 4, 160, 424, 132; 0, 0, 0, 1, 250, 1978, 2382, 429; 0, 0, 0, 0, 302, 6276, 19508, 12804, 1430; ... in which the main diagonal is the Catalan numbers, and the row sums form the factorials.
b:= proc(n) option remember; `if`(n=0, x,
expand((x-> x+x^2)(b(n-1))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=0..5); # Alois P. Heinz, Mar 14 2016
f[0][x_] = x; f[n_][x_] := f[n][x] = f[n-1][x+x^2]; row[n_] := CoefficientList[f[n][x], x] // Rest; Table[row[n], {n, 0, 5} ] // Flatten (* Jean-François Alcover, Sep 10 2012 *)
T(m,n):=if m=0 and n=1 then 1 else if m=0 and n>1 then 0 else if m=1 then binomial(1,n-1) else sum(binomial(i,n-i)*T(m-1,i),i,1,n); /* Vladimir Kruchinin, May 19 2012 */
{T(n,k)=local(F=x+x^2, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}
for(n=0, 6, for(k=1, 2^n, print1(T(n, k), ", ")); print(""))
Triangle begins: 1; 1, 1; 1, 1, 1; 1, 2, 1, 1; 1, 3, 3, 1, 1; 1, 6, 6, 4, 1, 1; 1, 12, 15, 10, 5, 1, 1; 1, 27, 40, 29, 15, 6, 1, 1; 1, 67, 113, 93, 49, 21, 7, 1, 1; 1, 180, 348, 310, 180, 76, 28, 8, 1, 1; 1, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1; 1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45, 10, 1, 1; 1, 5721, 15203, 16912, 11683, 5858, 2323, 750, 209, 55, 11, 1, 1; 1, 20924, 60710, 71858, 52262, 27349, 11230, 3809, 1083, 274, 66, 12, 1, 1; ... ILLUSTRATE RECURRENCE: T(6,1) = T(5,1) + T(5,2) = 6 + 6 = 12; T(7,2) = T(6,2) + 2*T(6,3) + T(6,4) = 6 + 2*4 + 1 = 15; T(8,3) = T(7,3) + 3*T(7,4) + 3*T(7,5) + T(7,6) = 29 + 3*15 + 3*6 + 1 = 93. Note that column 1 equals A122889: [1,1,2,3,6,12,27,67,180,528,...] which is the antidiagonal sums of triangle A122888. RELATED TRIANGLE A122888 begins: 1; 1, 1; 1, 2, 2, 1; 1, 3, 6, 9, 10, 8, 4, 1; 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1; 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...; 1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; 1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...; 1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...; ... in which the g.f. of row n equals the n-th iteration of (x+x^2).
T(n, k)=if(n
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