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A107671
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Triangular matrix T, read by rows, that satisfies: T = D + SHIFT_LEFT(T^3) where SHIFT_LEFT shifts each row 1 place left and D is the diagonal matrix {1,2,3,...}.
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6
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| Matrix diagonalization method: define triangular matrix P by P(n, k) = ((n+1)^3)^(n-k)/(n-k)!, n>=k>=0 and diagonal matrix D(n, n) = n+1, then T is given by T = P^-1*D*P.
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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 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; ...
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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])}
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CROSSREFS
| Cf. A107667, A107672 (column 0), A107673, A107674 (matrix square), A107676 (matrix cube), A006690.
Sequence in context: A188898 A176860 A093082 * A154538 A154166 A010521
Adjacent sequences: A107668 A107669 A107670 * A107672 A107673 A107674
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 07 2005
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