A143866 - OEIS

%I #14 May 09 2021 10:02:46

%S 1,1,1,2,1,2,3,2,2,5,5,3,4,5,12,7,5,6,10,12,29,11,7,10,15,24,29,69,15,

%T 11,14,25,36,58,69,165,22,15,22,35,60,87,138,165,393,937,42,30,44,75,

%U 132,203,345,495,786,937,2233,56,42,60,110,180,319,483,825,1179,1874,2233,5322

%N Eigentriangle of A027293.

%C Left border = partition numbers, A000041 starting (1, 1, 2, 3, 5, 7, ...). Right border = INVERT transform of partition numbers starting (1, 1, 2, 5, 12, ...); with row sums the same sequence but starting (1, 2, 5, 12, ...). Sum of n-th row terms = rightmost term of next row.

%C For another definition of L-eigen-matrix of A027293 see A343234. - _Wolfdieter Lang_, Apr 16 2021

%F Triangle read by rows, A027293 * (A067687 * 0^(n-k)); 1 <= k <= n. (A067687 * 0^(n-k)) = an infinite lower triangular matrix with the INVERT transform of the partition function as the main diagonal: (1, 1, 2, 5, 12, 29, 69, 165, ...); and the rest zeros. Triangle A027293 = n terms of "partition numbers decrescendo"; by rows = termwise product of n terms of partition decrescendo and n terms of A027293: (1, 1, 2, 5, 12, 29, 69, 165, ...).

%e The triangle begins:

%e n \ k    1  2  3  4   5   6   7   8   9  10   11 ...

%e -------------------------------------------

%e 1:       1

%e 2:       1  1

%e 3:       2  1  2

%e 4:       3  2  2  5

%e 5:       5  3  4  5  12

%e 6:       7  5  6 10  12  29

%e 7:      11  7 10 15  24  29  69

%e 8:      15 11 14 25  36  58  69 165

%e 9:      22 15 22 35  60  87 138 165 393

%e 10:     30 22 30 55  84 145 207 330 393 937

%e 11:     42 30 44 75 132 203 345 495 786 937 2233

%e ... reformatted and extended by _Wolfdieter Lang_, May 02 2021

%e Row 4 = (3, 2, 2, 5) = termwise product of (3, 2, 1, 1) and (1, 1, 2, 5) = (3*1, 2*1, 1*2, 1*5).

%Y Cf. A027293, A067687, A000041, A343234.

%K nonn,easy,tabl

%O 1,4

%A _Gary W. Adamson_, Sep 04 2008