A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 25, 25, 12, 4, 1, 138, 138, 63, 19, 5, 1, 970, 970, 421, 113, 28, 6, 1, 8390, 8390, 3472, 832, 190, 38, 7, 1, 86796, 86796, 34380, 7420, 1560, 283, 50, 8, 1, 1049546, 1049546, 399463, 78406, 15250, 2502, 411, 63, 9, 1, 14563135, 14563135

Offset

0,4

Comments

Column 0 is A135881 which equals column 0 of triangle A135879 and also equals column 0 of triangle A135880. Compare to triangle A135879, which is generated by a complementary process. An interesting variant is square array A135876, in which column 0 equals the double factorials (A001147).

Links

Table of n, a(n) for n=0..56.

Example

Square array begins:
(1),1,(1),1,(1),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),16,17,18,(19),20,...;
(2),6,(12),19,(28),38,50,(63),77,93,(110),128,148,169,(191),214,...;
(6),25,(63),113,(190),283,411,(559),728,942,(1181),1446,1766,2116,...;
(25),138,(421),832,(1560),2502,3948,(5714),7830,10740,(14130),18036,...;
(138),970,(3472),7420,(15250),25990,44026,(67112),95918,138343,(189598),..;
(970),8390,(34380),78406,(174324),312667,(563287),897471,1329234,2003240,..;
(8390),86796,(399463),962750,(2291984),4295224,8168819,(13523882),20656067,.;
(86796),1049546,(5344770),13513589,(34169656),66534382,132787852,(227380975),.;
(1049546),14563135,(81097517),213885369,(570682050),1149537869,2395865161,..;
(14563135),228448504,(1377986373),3773851534,(10568874312),21945438536,...;
where terms in parenthesis are removed before taking partial sums.
For example, to generate row 2 from row 1, remove terms at positions
{[(m+3)^2/4-2], m>=0} = [0,2,4,7,10,14,18,23,28,34,...] to obtain:
[2, 4, 6,7, 9,10, 12,13,14, 16,17,18, 20,21,22,23, ...]
then take partial sums to get row 2:
[2, 6, 12,19, 28,38, 50,63,77, 93,110,128, 148,169,191,214, ...].
Repeating this process will generate all the rows of the triangle.
Triangle A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and is generated by matrix powers of itself.

Prog

(PARI) {T(n, k)=local(A=0, b=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==floor((b+3)^2/4)-2, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Crossrefs

Cf. A135881, A135879, A135880; variants: A135876, A125714.

Sequence in context: A162980 A162979 A094587 * A329154 A121284 A225112

Adjacent sequences: A135875 A135876 A135877 * A135879 A135880 A135881

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 14 2007

Status

approved