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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.
5
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).
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
Sequence in context: A162980 A162979 A094587 * A329154 A121284 A225112
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 14 2007
STATUS
approved