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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. 5
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, 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:

1;

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;

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: A127126 A161133 A112911 * A152400 A291978 A111548

Adjacent sequences:  A152402 A152403 A152404 * A152406 A152407 A152408

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 05 2008

STATUS

approved

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Last modified November 18 01:00 EST 2018. Contains 317279 sequences. (Running on oeis4.)