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A111544 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+3 of T), or [T^p](m,0) = p*T(p+m,p+3) for all m>=1 and p>=-3. 10
1, 1, 1, 5, 2, 1, 33, 9, 3, 1, 261, 57, 15, 4, 1, 2361, 441, 99, 23, 5, 1, 23805, 3933, 783, 165, 33, 6, 1, 263313, 39249, 7083, 1383, 261, 45, 7, 1, 3161781, 430677, 71415, 13083, 2361, 393, 59, 8, 1, 40907241, 5137641, 789939, 136863, 23805, 3861, 567, 75, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column 0 equals A111530 (related to log of factorial series). Column 3 (A111547) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111549.

LINKS

Table of n, a(n) for n=0..54.

Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.

FORMULA

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+2, 2)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+3, 2) = 3*T(n+1, 0), T(n+4, 4) = T(n+1, 0), for n>=0.

EXAMPLE

SHIFT_LEFT(column 0 of T^-3) = -3*(column 0 of T);

SHIFT_LEFT(column 0 of T^-2) = -2*(column 1 of T);

SHIFT_LEFT(column 0 of T^-1) = -1*(column 2 of T);

SHIFT_LEFT(column 0 of log(T)) = column 3 of T;

SHIFT_LEFT(column 0 of T^1) = 1*(column 4 of T);

where SHIFT_LEFT of column sequence shifts 1 place left.

Triangle T begins:

1;

1,1;

5,2,1;

33,9,3,1;

261,57,15,4,1;

2361,441,99,23,5,1;

23805,3933,783,165,33,6,1;

263313,39249,7083,1383,261,45,7,1;

3161781,430677,71415,13083,2361,393,59,8,1; ...

After initial term, column 2 is 3 times column 0.

Matrix inverse T^-1 = A111548 starts:

1;

-1,1;

-3,-2,1;

-15,-3,-3,1;

-99,-15,-3,-4,1;

-783,-99,-15,-3,-5,1;

-7083,-783,-99,-15,-3,-6,1; ...

where columns are all equal after initial terms;

compare columns of T^-1 to column 2 of T.

Matrix logarithm log(T) = A111549 is:

0;

1,0;

4,2,0;

23,6,3,0;

165,32,9,4,0;

1383,222,47,13,5,0;

13083,1824,321,70,18,6,0; ...

compare column 0 of log(T) to column 3 of T.

MATHEMATICA

T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[n == k, 1, If[n == k + 1, n, k T[n, k + 1] + Sum[T[j + 2, 2] T[n, j + k + 1], {j, 0, n - k - 1}]]]];

Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Aug 09 2018, from PARI *)

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, if(n==k+1, n, k*T(n, k+1)+sum(j=0, n-k-1, T(j+2, 2)*T(n, j+k+1)))))

CROSSREFS

Cf. A111545 (column 1), A111546 (column 2), A111547 (column 3), A111552 (row sums), A111548 (matrix inverse), A111549 (matrix log); related tables: A111528, A104980, A111536, A111553.

Sequence in context: A281890 A342381 A341723 * A109281 A133289 A107719

Adjacent sequences:  A111541 A111542 A111543 * A111545 A111546 A111547

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Aug 07 2005

STATUS

approved

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Last modified January 28 15:20 EST 2022. Contains 350657 sequences. (Running on oeis4.)