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A194710 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (10 + m).

%I A194710 #33 May 15 2025 22:11:33
%S A194710 42,15,27,10,14,18,5,10,10,17,4,5,8,10,15,2,5,4,8,9,14,2,2,4,5,7,9,13,
%T A194710 1,2,2,4,4,8,8,13,1,1,2,2,4,4,7,9,12,0,1,1,2,2,4,4,7,8,13,1,0,1,1,2,2,
%U A194710 4,4,7,8,12,0,1,0,1,1,2,2,4,4,7,8,12
%N A194710 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (10 + m).
%C A194710 Sub-triangle of A182703 and also of A194812. Note that the sum of row k is also the number of partitions of 10. For further information see A182703 and A135010.
%F A194710 T(k,m) = A182703(10+m,k), with T(k,m) = 0 if k > 10+m.
%F A194710 T(k,m) = A194812(10+m,k).
%F A194710 Beginning with row k=11 each row starts with (k-11) 0's and ends with the subsequence 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, the initial terms of A002865. - _Alois P. Heinz_, Feb 15 2012
%e A194710 Triangle begins:
%e A194710   42;
%e A194710   15, 27;
%e A194710   10, 14, 18;
%e A194710    5, 10, 10, 17;
%e A194710    4,  5,  8, 10, 15;
%e A194710    2,  5,  4,  8,  9, 14;
%e A194710    2,  2,  4,  5,  7,  9, 13;
%e A194710    1,  2,  2,  4,  4,  8,  8, 13;
%e A194710    1,  1,  2,  2,  4,  4,  7,  9, 12;
%e A194710    0,  1,  1,  2,  2,  4,  4,  7,  8, 13;
%e A194710    1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
%e A194710    0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
%e A194710    0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
%e A194710    0,  0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
%e A194710   ...
%e A194710 For k = 1 and m = 1; T(1,1) = 42 because there are 42 parts of size 1 in the last section of the set of partitions of 11, since 10 + m = 11, so a(1) = 42. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the last section of the set of partitions of 11, since 10 + m = 11, so a(2) = 15.
%Y A194710 Always the sum of row k = p(10) = A000041(10) = 42.
%Y A194710 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194709, this sequence.
%Y A194710 Cf. A002865, A135010, A138121, A194812.
%K A194710 nonn,tabl
%O A194710 1,1
%A A194710 _Omar E. Pol_, Feb 05 2012