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 starts: 1: 1; 2: 1, 1; 3: 2, 0, 1; 4: 2, 2, 0, 1; 5: 3, 2, 1, 0, 1; 6: 4, 3, 2, 1, 0, 1; 7: 5, 4, 3, 1, 1, 0, 1; 8: 6, 7, 3, 3, 1, 1, 0, 1; 9: 8, 8, 6, 3, 2, 1, 1, 0, 1; 10: 10, 12, 7, 5, 3, 2, 1, 1, 0, 1; 11: 12, 15, 11, 6, 5, 2, 2, 1, 1, 0, 1; 12: 15, 21, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1; 13: 18, 26, 20, 12, 9, 5, 4, 2, 2, 1, 1, 0, 1; 14: 22, 35, 25, 18, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1; ... In the partition 5+2+2+2+1+1, 2 is repeated 3 times, no part is repeated more than 3 times. From _Gary W. Adamson_, Mar 13 2010: (Start) First few rows of the array = ... 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... = p(x)/p(x^2) = A000009 1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, ... = p(x)/p(x^3) 1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, ... = p(x)/p(x^4) 1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ... = p(x)/p(x^5) 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ... = p(x)/p(x^6) ... Finally, taking finite differences from the top and deleting the first "1", we obtain triangle A091602 with row sums = A000041 starting with offset 1: 1; 1, 1; 2, 0, 1; 2, 2, 0, 1; 3, 2, 1, 0, 1; 4, 3, 2, 1, 0, 1; ... (End)
g:=sum(t^k*(product((1-x^((k+1)*j))/(1-x^j),j=1..50)-product((1-x^(k*j))/(1-x^j),j=1..50)),k=1..50): gser:=simplify(series(g,x=0,20)): for n from 1 to 13 do P[n]:=coeff(gser,x^n) od: for n from 1 to 13 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 30 2006 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i>n, 0, add(b(n-i*j, i+1, min(k, iquo(n-i*j, i+1))), j=0..min(n/i, k)))) end: T:= (n, k)-> b(n, 1, k) -`if`(k=0, 0, b(n, 1, k-1)): seq(seq(T(n, k), k=1..n), n=1..20); # Alois P. Heinz, Nov 27 2013
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i>n, 0, Sum[b[n-i*j, i+1, Min[k, Quotient[n-i*j, i+1]]], {j, 0, Min[n/i, k]}]]]; t[n_, k_] := b[n, 1, k] - If[k == 0, 0, b[n, 1, k-1]]; Table[t[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz's second Maple program *)
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