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A104980 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T), or [T^p](m,0) = p*T(p+m,p+1) for all m>=1 and p>=-1. 15
1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 21, 5, 1, 3447, 1297, 461, 133, 31, 6, 1, 29093, 10063, 3447, 977, 225, 43, 7, 1, 273343, 87669, 29093, 8135, 1859, 353, 57, 8, 1, 2829325, 847015, 273343, 75609, 17185, 3251, 523, 73, 9, 1, 31998903 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column 0 equals A003319 (indecomposable permutations). Amazingly, column 1 (A104981) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A104986.

From Paul D. Hanna, Feb 17 2009: (Start)

Square array A156628 has columns found in this triangle T:

Column 0 of A156628 = column 0 of T = A003319;

Column 1 of A156628 = column 1 of T = A104981;

Column 2 of A156628 = column 2 of T = A003319 shifted;

Column 3 of A156628 = column 1 of T^2 (A104988);

Column 5 of A156628 = column 2 of T^2 (A104988). (End)

LINKS

Table of n, a(n) for n=0..55.

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

EXAMPLE

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

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

SHIFT_LEFT(column 0 of T^2) = 2*(column 3 of T);

where SHIFT_LEFT of column sequence shifts 1 place left.

Triangle T begins:

1;

1,1;

3,2,1;

13,7,3,1;

71,33,13,4,1;

461,191,71,21,5,1;

3447,1297,461,133,31,6,1;

29093,10063,3447,977,225,43,7,1;

273343,87669,29093,8135,1859,353,57,8,1;

2829325,847015,273343,75609,17185,3251,523,73,9,1; ...

Matrix inverse T^-1 is A104984 which begins:

1;

-1,1;

-1,-2,1;

-3,-1,-3,1;

-13,-3,-1,-4,1;

-71,-13,-3,-1,-5,1;

-461,-71,-13,-3,-1,-6,1; ...

Matrix T also satisfies:

[I + SHIFT_LEFT(T)] = [I - SHIFT_DOWN(T)]^-1, which starts:

1;

1,1;

2,1,1;

7,3,1,1;

33,13,4,1,1;

191,71,21,5,1,1; ...

where SHIFT_DOWN(T) shifts columns of T down 1 row,

and SHIFT_LEFT(T) shifts rows of T left 1 column,

with both operations leaving zeros in the diagonal.

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, 0] 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, 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, 0)*T(n, j+k+1)))))}

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

(PARI) {T(n, k) = if(n<k||k<0, 0, (matrix(n+1, n+1, m, j, if(m==j, 1, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x+O(x^m), m-j-1))))^-1)[n+1, k+1])}

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

CROSSREFS

Cf. A003319 (column 0), A104981 (column 1), A104983 (row sums), A104984 (matrix inverse), A104988 (matrix square), A104990 (matrix cube), A104986 (matrix log), A156628.

Sequence in context: A180190 A059438 A156628 * A316566 A134090 A132845

Adjacent sequences:  A104977 A104978 A104979 * A104981 A104982 A104983

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 10 2005

STATUS

approved

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Last modified October 17 08:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)