A156629 -id:A156629 - cp's OEIS frontend

A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 20, 5, 1, 3447, 1297, 461, 120, 28, 6, 1, 29093, 10063, 3447, 836, 181, 38, 7, 1, 273343, 87669, 29093, 6616, 1333, 270, 49, 8, 1, 2829325, 847015, 273343, 58576, 11029, 2150, 375, 61, 9, 1

Offset: 0

Paul D. Hanna, Feb 17 2009

			To generate the array, start with all 1's in row 0; from then on,
obtain row n+1 from row n by first removing terms in row n at
positions 0 and {(m+1)*(m+2)/2-2,m>0} and then taking partial sums.
This square array A begins:
(1), (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, 1, 1, 1, 1, ...;
(1), (2), 3, 4, (5), 6, 7, 8, (9), 10, 11, 12, 13, (14), 15, 16, ...;
(3), (7), 13, 20, (28), 38, 49, 61, (74), 89, 105, 122, 140, (159),...;
(13), (33), 71, 120, (181), 270, 375, 497, (637), 817, 1019, 1244, ...;
(71), (191), 461, 836, (1333), 2150, 3169, 4413, (5906), 8001, ...;
(461), (1297), 3447, 6616, (11029), 19030, 29483, 42775, (59324),...;
(3447), (10063), 29093, 58576, (101351), 185674, 300329, 451277, ...;
(29093), (87669), 273343, 573672, (1024949), 1982310, 3330651, ...;
(273343), (847015), 2829325, 6159976, (11320359), 23009602, 39998897, ...;
where terms in parenthesis at positions {0,1,4,8,13,..} in a row
are removed before taking partial sums to obtain the next row.
...
RELATION TO SPECIAL TRIANGLE.
Triangle A104980 begins:
1;
1, 1;
3, 2, 1;
13, 7, 3, 1;
71, 33, 13, 4, 1;
461, 191, 71, 21, 5, 1;
3447, 1297, 461, 133, 31, 6, 1;
29093, 10063, 3447, 977, 225, 43, 7, 1; ...
in which column 0 and column 1 are found in square array A.
...
Matrix square of A104980 = triangle A104988 which begins:
1;
2, 1;
8, 4, 1;
42, 20, 6, 1;
266, 120, 38, 8, 1;
1954, 836, 270, 62, 10, 1;
16270, 6616, 2150, 516, 92, 12, 1;
151218, 58576, 19030, 4688, 882, 128, 14, 1; ...
where column 1 and column 2 are also found in square array A.

Cf. columns: A003319, A104981, A156629, related triangles: A104980, A104988.

Cf. related tables: A136212, A136213, A125714, A135876, A127054, A125781, A136217.

{T (n, k)=local (A=0, b=2, c=1, d=0); if (n==0, A=1, until (d>k, if (c==b* (b+1)/2-2, b+=1, A+=T (n-1, c); d+=1); c+=1)); A}

Column 0 = Column 0 of triangle A104980 = A003319.
Column 1 = Column 1 of triangle A104980 = A104981.
Column 3 = column 1 of A104988 (matrix square of A104980).
Column 5 = column 2 of A104988 (matrix square of A104980).

A193630 Augmentation of the triangle A074909. See Comments.

Original entry on oeis.org

1, 1, 2, 1, 5, 7, 1, 9, 28, 33, 1, 14, 74, 181, 191, 1, 20, 159, 637, 1333, 1297, 1, 27, 300, 1767, 5906, 11029, 10063, 1, 35, 517, 4190, 20256, 59324, 101351, 87669, 1, 44, 833, 8873, 58339, 244125, 645146, 1024949, 847015, 1, 54, 1274, 17241, 147680

Offset: 0

Clark Kimberling, Aug 01 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193630, writing the general term as w(n,k),
w(n,n):  A104981
w(n,n-1):  A156629

			First five rows of A193607:
1
1...2
1...5....7
1...9....28...33
1...14...74...181...191

Cf. A193091, A074909.

p[n_, k_] := Binomial[n + 1, k];
Table[p[n, k], {n, 0, 7}, {k, 0, n}]  (* A074909 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193630 *)
Flatten[Table[v[n], {n, 0, 8}]]

cp's OEIS Frontend

A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

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