A073714 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.

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

Offset

0,5

Links

Table of n, a(n) for n=0..75.

Formula

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.

Example

Table read by antidiagonals gives first row; subsequent rows generated by convolutions of first row sequence.
1, 1,  1,   1,   2,   1,   1,   3,   3,   1,   1,   4,...;
1, 2,  3,   4,   7,   8,   9,  14,  20,  20,  21,  32,...;
1, 3,  6,  10,  18,  27,  37,  54,  81, 106, 132, 180,...;
1, 4, 10,  20,  39,  68,  54, 168, 263, 388, 544, 768,...;
1, 5, 15,  35,  75, 108, 260, 440, 730,1165,1781,2670,...;
1, 6, 21,  56, 132, 282, 552,1014,1794,3058,5013,...;
1, 7, 28,  84, 217, 504,1071,2122,4004,7252,...;
1, 8, 36, 120, 338, 848,1940,4120,8271,...;
1, 9, 45, 165, 504,1359,3327,7533,...;
1,10, 55, 220, 725,2092,5455,...;
1,11, 66, 286,1012,3113,...;
1,12, 78, 364,1377,...;
1,13, 91, 455,...;
1,14,105,...;
1,15,...; ...

Mathematica

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 *)

Crossrefs

Sequence in context: A118687 A281587 A026022 * A171848 A144151 A022818

Adjacent sequences: A073711 A073712 A073713 * A073715 A073716 A073717

Keyword

easy,nice,nonn,tabl

Author

Paul D. Hanna, Aug 05 2002

Extensions

a(65) corrected by Jean-François Alcover, Aug 05 2013

Status

approved