A125781 - OEIS

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A125781

Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 - 2 (m>=2).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 24, 23, 14, 5, 1, 1, 77, 76, 52, 21, 6, 1, 1, 295, 294, 217, 91, 29, 7, 1, 1, 1329, 1328, 1033, 433, 141, 39, 8, 1, 1, 6934, 6933, 5604, 2307, 739, 216, 50, 9, 1, 1, 41351, 41350, 34416, 13804, 4276, 1274, 306, 62, 10, 1, 1

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

OFFSET

0,5

COMMENTS

Generated by a method similar to Moessner's factorial triangle (A125714).

LINKS

Table of n, a(n) for n=0..66.

FORMULA

Surprisingly, column 1 equals A091352 = column 1 of triangle A091351, in which column k equals row sums of the matrix power A091351^k. Column 3 of this table also equals column 1 of matrix power A091351^2.

EXAMPLE

Rows are partial sums excluding terms in columns k = {1,4,8,13,...}:

row 2 = partial sums of [1, 3,4, 6,7,8, 10,11,12,13, ...];

row 3 = partial sums of [1, 8,14, 29,39,50, 75,90,106,123, ...];

row 4 = partial sums of [1, 23,52, 141,216,306, 535,695,876,1079,...].

The terms that are excluded in the partial sums are shown enclosed in

parenthesis in the table below. Rows of this table begin:

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, 17, ...;

1,(4), 8, 14,(21), 29, 39, 50,(62), 75, 90, 106, 123,(141), 160, 181,.;

1,(9), 23, 52,(91), 141, 216, 306,(412), 535, 695, 876, 1079,(1305),..;

1,(24), 76, 217,(433), 739, 1274, 1969,(2845), 3924, 5479, 7335,...;

1,(77), 294, 1033,(2307), 4276, 8200, 13679,(21014), 30534, 45528,...;

1,(295), 1328, 5604,(13804), 27483, 58017, 103545,(167868), 255305,...;

1,(1329), 6933, 34416,(92433), 195978, 451283, 855463,(1454823),...;

1,(6934), 41350, 237328,(688611), 1544074, 3847960, 7700971,...;

1,(41351), 278679, 1822753,(5670713), 13371684, 35818351, 75299744,...;

1,(278680), 2101433, 15473117,(51291468), 126591212, 362337006,...;

1,(2101434), 17574551, 144165763,(506502769), 1303252476,...;

1,(17574552), 161740315, 1464992791,(5430460072), 14517950305,...;

Column 1 of this table equals column 1 of triangle A091351;

triangle A091351 begins:

1, 1;

1, 2, 1;

1, 4, 3, 1;

1, 9, 9, 4, 1;

1, 24, 30, 16, 5, 1;

1, 77, 115, 70, 25, 6, 1;

1, 295, 510, 344, 135, 36, 7, 1;

1, 1329, 2602, 1908, 805, 231, 49, 8, 1; ...

where column k of A091351 = row sums of matrix power A091351^k for k>=0.

PROG

(PARI) {T(n, k)=local(A=0, b=2, c=0, 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}

CROSSREFS

Cf. A091351, A091352; columns: A125782, A125783, A125784, A125785, A125786; diagonals: A125787, A125788; A125789 (antidiagonal sums), A125714.

Sequence in context: A091491 A117418 A101494 * A091150 A091351 A058730

Adjacent sequences: A125778 A125779 A125780 * A125782 A125783 A125784

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 09 2006

STATUS

approved