0,4

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).

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.

(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}

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

Sequence in context: A137381 A109316 A094587 this_sequence A121284 A108076 A104557

Adjacent sequences: A135875 A135876 A135877 this_sequence A135879 A135880 A135881

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 14 2007