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A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+2 of T), or [T^p](m,0) = p*T(p+m,p+2) for all m>=1 and p>=-2. 9
1, 1, 1, 4, 2, 1, 22, 8, 3, 1, 148, 44, 14, 4, 1, 1156, 296, 84, 22, 5, 1, 10192, 2312, 600, 148, 32, 6, 1, 99688, 20384, 4908, 1156, 242, 44, 7, 1, 1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1, 12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column 0 equals A111529 (related to log of factorial series).

Column 2 (A111538) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111541.

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+1, 1)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+2, 1) = 2*T(n+1, 0), T(n+3, 3) = T(n+1, 0), for n>=0.

EXAMPLE

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

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

SHIFT_LEFT(column 0 of log(T)) = column 2 of T;

SHIFT_LEFT(column 0 of T^1) = 1*(column 3 of T);

SHIFT_LEFT(column 0 of T^2) = 2*(column 4 of T);

where SHIFT_LEFT of column sequence shifts 1 place left.

Triangle T begins:

1;

1, 1;

4, 2, 1;

22, 8, 3, 1;

148, 44, 14, 4, 1;

1156, 296, 84, 22, 5, 1;

10192, 2312, 600, 148, 32, 6, 1;

99688, 20384, 4908, 1156, 242, 44, 7, 1;

1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1;

12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1; ...

...

After initial term, column 1 is twice column 0.

Matrix inverse T^-1 = A111540 starts:

1;

-1, 1;

-2, -2, 1;

-8, -2, -3, 1;

-44, -8, -2, -4, 1;

-296, -44, -8, -2, -5, 1;

-2312, -296, -44, -8, -2, -6, 1;

-20384, -2312, -296, -44, -8, -2, -7, 1;

-199376, -20384, -2312, -296, -44, -8, -2, -8, 1; ...

where columns are all equal after initial terms;

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

Matrix logarithm log(T) = A111541 is:

0;

1, 0;

3, 2, 0;

14, 5, 3, 0;

84, 22, 8, 4, 0;

600, 128, 36, 12, 5, 0;

4908, 896, 212, 58, 17, 6, 0;

44952, 7220, 1496, 360, 90, 23, 7, 0;

454344, 65336, 12128, 2652, 602, 134, 30, 8, 0;

5016768, 653720, 110288, 22320, 4736, 974, 192, 38, 9, 0; ...

compare column 0 of log(T) to column 2 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+1, 1]*T[n, j+k+1], {j, 0, n-k-1}]]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 24 2017, adapted 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+1, 1)*T(n, j+k+1)))))

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A111537 (column 1), A111538 (column 2), A111539 (row sums), A111540 (matrix inverse), A111541 (matrix log); related tables: A111528, A104980, A111544, A111553.

Sequence in context: A152391 A144088 A039948 * A111559 A224798 A239894

Adjacent sequences:  A111533 A111534 A111535 * A111537 A111538 A111539

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Aug 06 2005

STATUS

approved

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Last modified January 16 19:48 EST 2022. Contains 350376 sequences. (Running on oeis4.)