A111554 -id:A111554 - cp's OEIS frontend

A111553 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+4 of T), or [T^p](m,0) = pT(p+m,p+4) for all m>=1 and p>=-4.

1, 1, 1, 6, 2, 1, 46, 10, 3, 1, 416, 72, 16, 4, 1, 4256, 632, 116, 24, 5, 1, 48096, 6352, 1016, 184, 34, 6, 1, 591536, 70912, 10176, 1664, 282, 46, 7, 1, 7840576, 864192, 113216, 17024, 2696, 416, 60, 8, 1, 111226816, 11371072, 1375456, 192384, 28792, 4256, 592, 76, 9, 1

Offset: 0

Paul D. Hanna, Aug 07 2005

Column 0 equals A111531 (related to log of factorial series). Column 4 (A111557) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111560.

			SHIFT_LEFT(column 0 of T^-4) = -4*(column 0 of T);
SHIFT_LEFT(column 0 of T^-3) = -3*(column 1 of T);
SHIFT_LEFT(column 0 of T^-2) = -2*(column 2 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 3 of T);
SHIFT_LEFT(column 0 of log(T)) = column 4 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 5 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1,1;
6,2,1;
46,10,3,1;
416,72,16,4,1;
4256,632,116,24,5,1;
48096,6352,1016,184,34,6,1;
591536,70912,10176,1664,282,46,7,1;
7840576,864192,113216,17024,2696,416,60,8,1; ...
After initial term, column 3 is 4 times column 0.
Matrix inverse T^-1 = A111559 starts:
1;
-1,1;
-4,-2,1;
-24,-4,-3,1;
-184,-24,-4,-4,1;
-1664,-184,-24,-4,-5,1;
-17024,-1664,-184,-24,-4,-6,1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 3 of T.
Matrix logarithm log(T) = A111560 is:
0;
1,0;
5,2,0;
34,7,3,0;
282,44,10,4,0;
2696,354,60,14,5,0;
28792,3328,470,84,19,6,0; ...
compare column 0 of log(T) to column 4 of T.

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

Cf. A111531 (column 0), A111554 (column 1), A111555 (column 2), A111556 (column 3), A111557 (column 4), A111558 (row sums), A111559 (matrix inverse), A111560 (matrix log); related tables: A111528, A104980, A111536, A111544.

T[n_, k_] := T[n, k] = If[nJean-François Alcover, Aug 09 2018, from PARI *)

PARI
```
T(n,k)=if(n
				
```

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+3, 3)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+4, 3) = 4*T(n+1, 0), T(n+5, 5) = T(n+1, 0), for n>=0.

A245892 Number of labeled increasing unary-binary trees on n nodes whose breadth-first reading word simultaneously avoids 231 and 312 (or avoids 312 and 321).

Original entry on oeis.org

1, 1, 3, 8, 28, 102, 407, 1701

Offset: 1

Manda Riehl, Aug 19 2014

The number of labeled increasing unary-binary trees with an associated permutation avoiding 231 and 312 simultaneously (or avoids 312 and 321 simultaneously) in the classical sense. The tree's permutation is found by recording the labels in the order in which they appear in a breadth-first search. (Note that a breadth-first search reading word is equivalent to reading the tree labels left to right by levels, starting with the root.)

In some cases, the same breadth-first search reading permutation can be found on differently shaped trees. This sequence gives the number of trees, not the number of permutations.

			When n=4, a(n)=8.  In the Links above we show the eight labeled increasing trees on four nodes whose permutation simultaneously avoids 231 and 312.

Manda Riehl, The eight trees when n=4.

A111554 gives the number of binary trees instead of unary-binary trees.

A002083 gives the number of permutations which avoid 231 and 312 simultaneously that are breadth-first reading words on labeled increasing unary-binary trees.

cp's OEIS Frontend

A111553 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+4 of T), or [T^p](m,0) = pT(p+m,p+4) for all m>=1 and p>=-4.

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A245892 Number of labeled increasing unary-binary trees on n nodes whose breadth-first reading word simultaneously avoids 231 and 312 (or avoids 312 and 321).

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cp's OEIS Frontend

A111553 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+4 of T), or [T^p](m,0) = p*T(p+m,p+4) for all m>=1 and p>=-4.

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Mathematica

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Formula

A245892 Number of labeled increasing unary-binary trees on n nodes whose breadth-first reading word simultaneously avoids 231 and 312 (or avoids 312 and 321).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A111553 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+4 of T), or [T^p](m,0) = pT(p+m,p+4) for all m>=1 and p>=-4.