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A136221 Column 0 of triangles A136220 and A136228; also equals column 0 of tables A136217 and A136218. 11
1, 1, 3, 15, 108, 1036, 12569, 185704, 3247546, 65762269, 1515642725, 39211570981, 1125987938801, 35554753133312, 1224882431140838, 45731901253649898, 1839804317195739634, 79355626796692509253, 3653687500034925338348 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

P = A136220 is a triangular matrix where column k of P^3 equals column 0 of P^(3k+3) such that column 0 of P^3 equals column 0 of P shift one place left. Tables A136217 and A136218 are defined by recurrences seemingly unrelated to the matrix product recurrence of A136220 and yet they all generate this same sequence in column 0.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..90

EXAMPLE

Equals column 0 of triangle P=A136220, which begins:

1;

1, 1;

3, 2, 1;

15, 10, 3, 1;

108, 75, 21, 4, 1;

1036, 753, 208, 36, 5, 1;

12569, 9534, 2637, 442, 55, 6, 1;

185704, 146353, 40731, 6742, 805, 78, 7, 1; ...

where column k of P^3 = column 0 of P^(3k+3) such that

column 0 of P^3 = column 0 of P shift one place left.

Surprisingly, column 0 of P is also found in square A136218:

(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),...;

(3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),...;

(15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),...;

(108),(414),1036,(2116),3493,(5555),8040,(11477),15483,...;

(1036),(4529),12569,(28052),48800,(82328),124335,(186261),...;

(12569),(61369),185704,(446560),811111,(1438447),2250731,...;

...

and has a recurrence similar to that of square array A136212

which generates the triple factorials.

PROG

(PARI) /* Generate using matrix product recurrences of triangle A136220: */ {a(n)=local(P=Mat(1), U, PShR); if(n>0, for(i=0, n, PShR=matrix(#P, #P, r, c, if(r>=c, if(r==c, 1, if(c==1, 0, P[r-1, c-1])))); U=P*PShR^2; U=matrix(#P+1, #P+1, r, c, if(r>=c, if(r<#P+1, U[r, c], if(c==1, (P^3)[ #P, 1], (P^(3*c-1))[r-c+1, 1])))); P=matrix(#U, #U, r, c, if(r>=c, if(r<#R, P[r, c], (U^c)[r-c+1, 1]))))); P[n+1, 1]}

(PARI) /* Generated as column 0 in triangle A136218 (faster): */ {a(n)=local(A=[1], B); if(n>0, for(i=1, n, m=1; B=[0]; for(j=1, #A, if(j+m-1==(m*(m+7))\6, m+=1; B=concat(B, 0)); B=concat(B, A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B))))))); A[1]}

CROSSREFS

Cf. A136220 (P), A136228 (U), A136231 (W=P^3).

Cf. other columns of P: A136222, A136223, A136224.

Cf. related tables: A136217, A136218.

Cf. variants: A091352, A135881.

Sequence in context: A120732 A245835 A090351 * A153305 A110328 A217061

Adjacent sequences:  A136218 A136219 A136220 * A136222 A136223 A136224

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 25 2007

STATUS

approved

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Last modified September 22 05:11 EDT 2021. Contains 347605 sequences. (Running on oeis4.)