A152405 - OEIS

A152405

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

1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 86, 45, 14, 4, 1, 645, 318, 86, 22, 5, 1, 5662, 2671, 645, 152, 31, 6, 1, 56632, 25805, 5662, 1251, 232, 41, 7, 1, 633545, 280609, 56632, 11869, 2026, 327, 53, 8, 1, 7820115, 3381993, 633545, 126987, 20143, 2991, 457, 66, 9, 1

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

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..54.

EXAMPLE

Table begins:

(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),(8),14,(22),31,41,(53),66,80,95,(112),130,149,169,190,...;

(14),(45),86,(152),232,327,(457),606,775,965,(1202),1464,1752,2067,...;

(86),(318),645,(1251),2026,2991,(4455),6207,8274,10684,(13934),17653,...;

(645),(2671),5662,(11869),20143,30827,(48480),70355,96990,128959,...;

(5662),(25805),56632,(126987),223977,352936,(582183),874664,1240239,...;

(56632),(280609),633545,(1508209),2748448,4438122,(7641111),11831184,...;

(633545),(3381993),7820115,(19651299),36837937,60743909,...; ...

where row n equals the partial sums of row n-1 after removing terms

at positions {m*(m+1)/2, m>=0} (marked by parenthesis in above table).

For example, to generate row 3 from row 2:

[3,8, 14, 22, 31,41, 53, 66,80,95, 112, 130,...]

remove terms at positions {0,1,3,6,10,...}, yielding:

[14, 31,41, 66,80,95, 130,149,169,190, ...]

then take partial sums to obtain row 3:

[14, 45,86, 152,232,327, 457,606,775,965, ...].

Continuing in this way generates all rows of this table.

RELATION TO POWERS OF A SPECIAL TRIANGULAR MATRIX.

Columns 0 and 1 are found in triangle T=A152400, which begins:

1, 1;

3, 2, 1;

14, 8, 3, 1;

86, 45, 15, 4, 1;

645, 318, 99, 24, 5, 1;

5662, 2671, 794, 182, 35, 6, 1;

56632, 25805, 7414, 1636, 300, 48, 7, 1; ...

where column k of T = column 0 of matrix power T^(k+1) for k>=0.

Furthermore, matrix powers of triangle T=A152400 satisfy:

column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

Column 3 of this square array = column 1 of T^2:

2, 1;

8, 4, 1;

45, 22, 6, 1;

318, 152, 42, 8, 1;

2671, 1251, 345, 68, 10, 1;

25805, 11869, 3253, 648, 100, 12, 1; ...

RELATED TRIANGLE A127714 begins:

1, 1, 1;

1, 2, 2, 3, 3, 3;

1, 3, 5, 5, 8, 11, 11, 14, 14, 14;

1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...

where right border = column 0 of this square array.

PROG

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

CROSSREFS

Cf. columns: A127715, A152401, A152404.

Cf. related triangles: A152400, A127714.

Cf. variants: A125781, A127054, A136212, A136217, A135876, A135878, A136730.

Sequence in context: A371461 A161133 A112911 * A152400 A291978 A342217

Adjacent sequences: A152402 A152403 A152404 * A152406 A152407 A152408

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 05 2008

STATUS

approved