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A177390 Triangle T where the g.f. of row n of T^(2n) = (2n^2 + y)^n for n>=0, as read by rows, where T^n denotes the n-th matrix power of T. 5
1, 1, 1, 10, 4, 1, 447, 72, 9, 1, 50040, 4624, 264, 16, 1, 10435970, 683300, 23750, 700, 25, 1, 3470932404, 178979256, 4569480, 84840, 1530, 36, 1, 1677020809366, 72215891104, 1489987002, 20776980, 241325, 2940, 49, 1, 1106343610197376 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Analogous to Pascal's triangle, C, which obeys a similar rule: the g.f. of row n of C^(2n) = (2n + y)^n for n>=0.

Conjecture: for all integer k, there exists an integer triangle J such that the g.f. of row n of J^(k*n) = (k*n^2 + y)^n for n>=0.

LINKS

Table of n, a(n) for n=0..36.

EXAMPLE

Triangle T begins:

1;

1,1;

10,4,1;

447,72,9,1;

50040,4624,264,16,1;

10435970,683300,23750,700,25,1;

3470932404,178979256,4569480,84840,1530,36,1;

1677020809366,72215891104,1489987002,20776980,241325,2940,49,1;

1106343610197376,41253720775296,725138126272,8309193088,73585120,586432,5152,64,1;

953498812570622640,31544658525648240,487943071058088,4827635270640,35544216204,218340360,1269324,8424,81,1;

...

Matrix square T^2 begins:

1;

2,1; <== (2 + y)^1 = g.f. for row 1 of T^2

24,8,1;

1056,180,18,1;

114496,11456,672,32,1;

23356640,1627600,60400,1800,50,1;

...

Matrix power T^4 begins:

1;

4,1;

64,16,1; <== (2*2^2 + y)^2 = g.f. for row 2 of T^4

2904,504,36,1;

301824,34048,1920,64,1;

59043680,4635200,186800,5200,100,1;

...

Matrix power T^6 begins:

1;

6,1;

120,24,1;

5832,972,54,1; <== (2*3^2 + y)^3 = g.f. for row 3 of T^6

598080,72384,3744,96,1;

113094720,9838800,408000,10200,150,1;

...

Matrix power T^8 begins:

1;

8,1;

192,32,1;

10128,1584,72,1;

1048576,131072,6144,128,1; <== (2*4^2 + y)^4 = g.f. for row 4 of T^8

193866560,18284800,752800,16800,200,1;

...

PROG

(PARI) {T(n, k, p=2)=local(M=Mat(1), N, L); for(i=1, n, N=M; M=matrix(#N+1, #N+1, r, c, if(r>=c, if(r<=#N, (N^(p*(#N)))[r, c], polcoeff((x+p*(#M)^2)^(#M), c-1)))); L=sum(i=1, #M, -(M^0-M)^i/i); M=sum(i=0, #M, (L/p/(#N))^i/i!); ); M[n+1, k+1]}

CROSSREFS

Cf. columns: A177391, A177392, A177393, row sums: A177394.

Cf. variant: A132870.

Sequence in context: A028967 A097530 A063565 * A082961 A065194 A113315

Adjacent sequences:  A177387 A177388 A177389 * A177391 A177392 A177393

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, May 25 2010

STATUS

approved

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Last modified September 17 16:57 EDT 2019. Contains 327136 sequences. (Running on oeis4.)