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.
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
Examples
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.
Links
- Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
Crossrefs
Programs
-
Mathematica
T[n_, k_] := T[n, k] = If[n
Jean-François Alcover, Aug 09 2018, from PARI *) -
PARI
T(n,k)=if(n
Formula
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.
Comments