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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 25 22:17 EDT 2019. Contains 323576 sequences. (Running on oeis4.)