A073714 - OEIS

%I #17 Aug 05 2013 07:09:09

%S 1,1,1,1,2,1,1,3,3,1,1,4,6,4,2,1,5,10,10,7,1,1,6,15,20,18,8,1,1,7,21,

%T 35,39,27,9,3,1,8,28,56,75,68,37,14,3,1,9,36,84,132,146,108,54,20,1,1,

%U 10,45,120,217,282,260,168,81,20,1,1,11,55,165,338,504,552,440,263,106

%N Table, read by antidiagonals, generated by successive convolutions of the first row such that the first row equals the same table in this flattened form.

%F Let first row sequence be a(n)=T(0, n); define f(x) = sum_{k=0..inf} a(k)x^k, then the n-th row is generated by: f(x)^(n+1) = sum_{k=0..inf} T(n, k)x^k.

%e Table read by antidiagonals gives first row; subsequent rows generated by convolutions of first row sequence.

%e 1, 1,  1,   1,   2,   1,   1,   3,   3,   1,   1,   4,...;

%e 1, 2,  3,   4,   7,   8,   9,  14,  20,  20,  21,  32,...;

%e 1, 3,  6,  10,  18,  27,  37,  54,  81, 106, 132, 180,...;

%e 1, 4, 10,  20,  39,  68,  54, 168, 263, 388, 544, 768,...;

%e 1, 5, 15,  35,  75, 108, 260, 440, 730,1165,1781,2670,...;

%e 1, 6, 21,  56, 132, 282, 552,1014,1794,3058,5013,...;

%e 1, 7, 28,  84, 217, 504,1071,2122,4004,7252,...;

%e 1, 8, 36, 120, 338, 848,1940,4120,8271,...;

%e 1, 9, 45, 165, 504,1359,3327,7533,...;

%e 1,10, 55, 220, 725,2092,5455,...;

%e 1,11, 66, 286,1012,3113,...;

%e 1,12, 78, 364,1377,...;

%e 1,13, 91, 455,...;

%e 1,14,105,...;

%e 1,15,...; ...

%t max = 75; a[0] = 1; se[n_] := se[n] = Series[ Sum[x^(j*(j + 1)/2)*(1 + x)^j, {j, 0, max - n}]^(n + 1), {x, 0, max - n}]; t[n_, k_] := t[n, k] = Coefficient[se[n], x, k]; ft = Flatten[ Table[t[n - j, j], {n, 0, max}, {j, 0, n}]][[1 ;; max + 1]]; sol = Thread[ft == Table[a[k], {k, 0, max}]] // Solve; sol /. Rule -> Set; Table[a[k], {k, 0, max}] (* _Jean-François Alcover_, Aug 05 2013 *)

%K easy,nice,nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Aug 05 2002

%E a(65) corrected by _Jean-François Alcover_, Aug 05 2013