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A107671 Triangular matrix T, read by rows, that satisfies: T = D + SHIFT_LEFT(T^3), where SHIFT_LEFT shifts each row 1 place to the left and D is the diagonal matrix {1, 2, 3, ...}. 6
1, 8, 2, 513, 27, 3, 81856, 2368, 64, 4, 23846125, 469625, 7625, 125, 5, 10943504136, 160767720, 1898856, 19656, 216, 6, 7250862593527, 83548607478, 776598305, 6081733, 43561, 343, 7, 6545029128786432, 61068815111168, 465690017280, 2966844928, 16494080, 86528, 512, 8 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..35.

FORMULA

Matrix diagonalization method: define the triangular matrix P by P(n, k) = ((n+1)^3)^(n-k)/(n-k)! for n >= k >= 0 and the diagonal matrix D(n, n) = n+1 for n >= 0; then T is given by T = P^-1*D*P.

T(n,k=0) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (s_1^(-1)/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(3*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n + 1. (Thus, the second sum is over all compositions of n + 1.) - Petros Hadjicostas, Mar 11 2021

EXAMPLE

Triangle T begins:

              1;

              8,           2;

            513,          27,         3;

          81856,        2368,        64,       4;

       23846125,      469625,      7625,     125,     5;

    10943504136,   160767720,   1898856,   19656,   216,   6;

  7250862593527, 83548607478, 776598305, 6081733, 43561, 343, 7;

  ...

The matrix cube T^3 shifts each row to the right 1 place, dropping the diagonal D and putting A006690 in column 0:

             1;

            56,           8;

          7965,         513,        27;

       2128064,       81856,      2368,      64;

     914929500,    23846125,    469625,    7625,  125;

  576689214816, 10943504136, 160767720, 1898856, 19656, 216;

  ...

From Petros Hadjicostas, Mar 11 2021: (Start)

We illustrate the above formula for T(n,k=0) with the compositions of n + 1 for n = 2. The compositions of n + 1 = 3 are 3, 1 + 2, 2 + 1, and 1 + 1 + 1.  Thus the above sum has four terms with (r = 1, s_1 = 3), (r = 2, s_1 = 1, s_2 = 2), (r = 2, s_1 = 2, s_2 = 1), and (r = 3, s_1 = s_2 = s_3 = 1).

The value of the denominator Product_{j=1..r} s_j! for these four terms is 6, 2, 2, and 1, respectively.

The value of the numerator s_1^(-1)*Product_{j=1..r} (Sum_{i=1..j} s_i)^(3*s_j) for these four terms is 19683/3, 729/1, 1728/2, and 216/1.

Thus T(2,0) = (19683/3)/6 - (729/1)/2 - (1728/2)/2 + (216/1)/1 = 513. (End)

PROG

(PARI) {T(n, k)=local(P=matrix(n+1, n+1, r, c, if(r>=c, (r^3)^(r-c)/(r-c)!)), D=matrix(n+1, n+1, r, c, if(r==c, r))); if(n>=k, (P^-1*D*P)[n+1, k+1])}

CROSSREFS

Cf. A107667, A107672 (column 0), A107673, A107674 (matrix square), A107676 (matrix cube), A006690.

Sequence in context: A176860 A281068 A093082 * A271174 A216891 A256783

Adjacent sequences:  A107668 A107669 A107670 * A107672 A107673 A107674

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jun 07 2005

STATUS

approved

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Last modified November 30 17:12 EST 2021. Contains 349424 sequences. (Running on oeis4.)