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A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.

%I A277627 #73 Sep 16 2021 02:43:11
%S A277627 0,0,1,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,0,2,1,0,0,0,0,0,3,1,0,0,0,0,0,1,
%T A277627 4,1,0,0,0,0,0,0,3,5,1,0,0,0,0,0,0,0,6,6,1,0,0,0,0,0,0,0,1,10,7,1,0,0,
%U A277627 0,0,0,0,0,0,4,15,8,1,0,0,0,0,0,0,0,0,0,10,21,9,1,0,0,0,0,0,0,0,0,0,1,20,28,10,1
%N A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.
%C A277627 In other words, for n > 0 the column k lists 2*k+1 zeros together with the partial sums of the positive terms of column k-1. - _Omar E. Pol_, Oct 25 2016
%C A277627 Comments from the author:
%C A277627 1) ZSPEC =
%C A277627   0, 0, 0, 0, 0, 0, 0, 0, ...
%C A277627   1, 0, 0, 0, 0, 0, 0, 0, ...
%C A277627   1, 0, 0, 0, 0, 0, 0, 0, ...
%C A277627   1, 1, 0, 0, 0, 0, 0, 0, ...
%C A277627   1, 2, 0, 0, 0, 0, 0, 0, ...
%C A277627   1, 3, 1, 0, 0, 0, 0, 0, ...
%C A277627   1, 4, 3, 0, 0, 0, 0, 0, ...
%C A277627   1, 5, 6, 1, 0, 0, 0, 0, ...
%C A277627 etc.
%C A277627 The columns are the autosequences of the first kind of the title (column 1: 0, 0, followed by A001477(n); column 2: 0, 0, 0, 0, followed by A000217(n), etc) .
%C A277627 The positive terms are the Pascal triangle written by diagonals (A011973).
%C A277627 First column: A060576(n+1). Or A057427(n), n>-1, thanks to Omar E. Pol.
%C A277627 Row sums: A000045(n), autosequence of the first kind.
%C A277627 Alternated row sums and subtractions: 0, 1, 1, 0, -1, -1, 0 = A128834(n), autosequence of the first kind.
%C A277627 Antidiagonal sums: 0, 1, 1, 1, 2, 3, 4, 6, ... = A078012(n+2).
%C A277627 Application.
%C A277627 Numbers in triangle leading to the Genocchi numbers -A226158(n).
%C A277627 We multiply the columns of ZSPEC by d(n) = 1, -1, 2, -8, 56, -608, ... from A005439.
%C A277627 Hence, with only the first 0,
%C A277627   0,
%C A277627   1,
%C A277627   1,
%C A277627   1, -1,
%C A277627   1, -2,
%C A277627   1, -3,  2,
%C A277627   1, -4,  6,
%C A277627   1, -5, 12,   -8,
%C A277627   1, -6, 20,  -32,
%C A277627   1, -7, 30,  -80,  56,
%C A277627   1, -8, 42, -160, 280,
%C A277627   etc.
%C A277627 The row sums is -A226158(n).
%C A277627 2) Now consider the case of the autosequences of the second kind.
%C A277627 First step.
%C A277627             2, 1, 1,  1,  1,  1, ...        = A054977(n)
%C A277627       0, 0, 2, 3, 4,  5,  6,  7, ...        = A199969(n) with offset 0
%C A277627 0, 0, 0, 0, 2, 5, 9, 14, 20, 27, ...        see A000096
%C A277627 etc.
%C A277627 The positive terms are ASPEC in A191302. By triangle, they are either A029653(n) with A029653(0) = 2 instead of 1 or A029635(n).
%C A277627 Second step. YSPEC =
%C A277627   2, 0,  0, 0, 0, 0, ...
%C A277627   1, 0,  0, 0, 0, 0, ...
%C A277627   1, 2,  0, 0, 0, 0, ...
%C A277627   1, 3,  0, 0, 0, 0, ...
%C A277627   1, 4,  2, 0, 0, 0, ...
%C A277627   1, 5,  5, 0, 0, 0, ...
%C A277627   1, 6,  9, 2, 0, 0, ...
%C A277627   1, 7, 14, 7, 0, 0, ...
%C A277627   etc.
%C A277627 Diagonals by triangle: A029635(n).
%C A277627 This is the companion to ZSPEC.
%C A277627 Row sums: A000032(n), autosequence of the second kind.
%C A277627 Alternated row sums and subtractions: period 6 repeat 2, 1, -1, -2, -1, 1 = A087204(n), autosequence of the second kind.
%C A277627 Application.
%C A277627 Numbers in triangle leading to A230324(n), a companion to -A226158(n).
%C A277627 We multiply the columns of YSPEC by d(n) 1, -1, 2, -8, 56, ... (see above).
%C A277627 Hence, without zeros:
%C A277627   2,
%C A277627   1,
%C A277627   1, -2,
%C A277627   1, -3,
%C A277627   1, -4,  4,
%C A277627   1, -5, 10,
%C A277627   1, -6, 18,  -16,
%C A277627   1, -7, 28,  -56,
%C A277627   1, -8, 40, -128, 112,
%C A277627   1, -9, 54, -240, 504,
%C A277627   etc.
%C A277627 The row sum is A230324(n).
%H A277627 G. C. Greubel, <a href="/A277627/b277627.txt">Table of n, a(n) for the first 25 antidiagonals, flattened</a>
%H A277627 OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%t A277627 kMax = 13; col[0] = Join[{0}, Array[1&, kMax]]; col[k_] := col[k] = Join[{0, 0}, col[k-1][[1 ;; -3]] // Accumulate]; T[n_, k_] := col[k][[n+1]]; Table[T[n-k, k], {n, 0, kMax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 15 2016 *)
%Y A277627 Cf. A011973 (without 0's), A007318 (Pascal's triangle).
%Y A277627 Cf. A000045 (row sums), A078012 (antidiagonal sums).
%Y A277627 Columns: A060576 or A057427 (k=0), A001477 (k=1), A000217 (k=2).
%Y A277627 Cf. A000004, A000012, A000027, A000032, A005439, A029635, A029653, A054977, A087204, A128834, A191302, A199969, A226158, A230324.
%K A277627 nonn,tabl
%O A277627 0,20
%A A277627 _Paul Curtz_, Oct 24 2016
%E A277627 Better definition from _Omar E. Pol_, Oct 25 2016