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%I #3 Mar 30 2012 18:36:58
%S 1,1,1,1,2,4,1,3,9,30,1,4,15,69,335,1,5,22,118,769,4984,1,6,30,178,
%T 1317,11346,92652,1,7,39,250,1995,19311,208914,2065146,1,8,49,335,
%U 2820,29126,352636,4613976,53636520,1,9,60,434,3810,41061,528097,7722840,118840164
%N Rectangular table, read by antidiagonals, where row n is equal to column 1 of matrix power A121412^(n+1) for n>=0.
%e Table of column 1 in matrix powers of triangle H=A121412 begins:
%e H^1: 1, 1, 4, 30, 335, 4984, 92652, 2065146, 53636520, ...
%e H^2: 1, 2, 9, 69, 769, 11346, 208914, 4613976, 118840164, ...
%e H^3: 1, 3, 15, 118, 1317, 19311, 352636, 7722840, 197354133, ...
%e H^4: 1, 4, 22, 178, 1995, 29126, 528097, 11476963, 291124693, ...
%e H^5: 1, 5, 30, 250, 2820, 41061, 740035, 15971180, 402319275, ...
%e H^6: 1, 6, 39, 335, 3810, 55410, 993678, 21310710, 533345745, ...
%e H^7: 1, 7, 49, 434, 4984, 72492, 1294776, 27611970, 686872893, ...
%e H^8: 1, 8, 60, 548, 6362, 92652, 1649634, 35003430, 865852191, ...
%e H^9: 1, 9, 72, 678, 7965, 116262, 2065146, 43626510, 1073540871, ...
%e Rearrangement of the upper part of the table forms A121431, which is
%e the number of subpartitions of partition [0,0,1,1,1,2,2,2,2,3,...]:
%e 1,1, 1,2,3, 4,9,15,22, 30,69,118,178,250, 335,769,1317,1995,2820,...
%o (PARI) {T(n,k)=local(H=Mat(1), B); for(m=1, k+2, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(H^i)[i-1, j]); )); H=B); return((H^(n+1))[k+2, 2])}
%Y Cf. A121427 (diagonal), A121431; rows: A121413, A121417, A121422; related tables: A121424, A121428; related triangles: A121412, A121416, A121420.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Aug 26 2006
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