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%I #3 Mar 30 2012 18:36:58
%S 1,1,1,1,2,3,1,3,7,18,1,4,12,43,170,1,5,18,76,403,2220,1,6,25,118,711,
%T 5188,37149,1,7,33,170,1107,9054,85569,758814,1,8,42,233,1605,13986,
%U 147471,1725291,18301950,1,9,52,308,2220,20171,225363,2938176,41145705
%N Rectangular table, read by antidiagonals, where row n is equal to column 0 of matrix power A121412^(n+1) for n>=0.
%e Table of column 0 in matrix powers of triangle H=A121412 begins:
%e H^1: 1, 1, 3, 18, 170, 2220, 37149, 758814, 18301950,...
%e H^2: 1, 2, 7, 43, 403, 5188, 85569, 1725291, 41145705,...
%e H^3: 1, 3, 12, 76, 711, 9054, 147471, 2938176, 69328365,...
%e H^4: 1, 4, 18, 118, 1107, 13986, 225363, 4441557, 103755660,...
%e H^5: 1, 5, 25, 170, 1605, 20171, 322075, 6285390, 145453290,...
%e H^6: 1, 6, 33, 233, 2220, 27816, 440785, 8526057, 195579123,...
%e H^7: 1, 7, 42, 308, 2968, 37149, 585046, 11226958, 255436293,...
%e H^8: 1, 8, 52, 396, 3866, 48420, 758814, 14459138, 326487241,...
%e H^9: 1, 9, 63, 498, 4932, 61902, 966477, 18301950, 410368743,...
%e Rearrangement of the upper half of the table forms A121430, which is
%e the number of subpartitions of partition [0,1,1,2,2,2,3,3,3,3,4,...]:
%e 1, 1,2, 3,7,12, 18,43,76,118, 170,403,711,1107,1605, 2220,...
%o (PARI) {T(n,k)=local(H=Mat(1), B); for(m=1, k+1, 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+1, 1])}
%Y Cf. A121425 (diagonal), A121430; rows: A101483, A121418, A121421; related tables: A121426, A121428; related triangles: A121412, A121416, A121420.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Aug 26 2006
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