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A277513 Irregular triangle read by rows: T(n,k) is the number of integers greater than 4 such that they have n trits and 2k+1 (k>=1) nonzero trits in their balanced ternary representation, with n>=3 and 1<=k<=(j-1)/2.

%I A277513 #34 Feb 23 2018 15:43:30
%S A277513 4,12,24,16,40,80,60,240,64,84,560,448,112,1120,1792,256,144,2016,
%T A277513 5376,2304,180,3360,13440,11520,1024,220,5280,29568,42240,11264,264,
%U A277513 7920,59136,126720,67584,4096,312,11440,109824,329472,292864,53248,364,16016,192192
%N A277513 Irregular triangle read by rows: T(n,k) is the number of integers greater than 4 such that they have n trits and 2k+1 (k>=1) nonzero trits in their balanced ternary representation, with n>=3 and 1<=k<=(j-1)/2.
%C A277513 This is a subset of A013609 and A188440.
%C A277513 This sequence T(n,k) can be re-indexed into the form of b(m) where m is positive integer sequence that can be calculated by parametric function m = m(n,k): m(n=2j+1,k) = j^2-j+k, where n is odd and 1<=k<=j.
%C A277513 m(n=2j,k) = j^2-2j+1+k, where n is even and 1<=k<=j.
%C A277513 Here n is the number of digits of an odd number in balanced ternary representation, and 2k+1 is the number of nonzero trits (1 or T) of the same number in balanced ternary representation.
%H A277513 Lei Zhou, <a href="/A277513/b277513.txt">Table of n, a(n) for n = 3..10002</a>
%H A277513 Definition of <a href="https://en.wikipedia.org/wiki/Balanced_ternary">Balanced Ternary</a>.
%F A277513 T(n,k) = 2^(2k)*Binomial(n-1, 2k)
%e A277513 Odd numbers that can be expressed in 3 trits balanced ternary (bt) form are 5 = 1TT, 7 = 1T1, 9 = 100, 11 = 11T, 13 = 111. Among these five numbers, four have 3 nonzero digits, so a(1) = 4.
%e A277513 Odd numbers in 4 trits bt form are 15 = 1TT0, 17 = 1T0T, 19 = 1T01, 21 = 1T10, 23 = 10TT, 25 = 10T1, 27 = 1000, 29 = 101T, 31 = 1011, 33 = 11T0, 35 = 110T, 37 = 1101, 39 = 1110. Among these 13 numbers, 12 have 3 nonzero digits, so a(2) = 12.
%e A277513 The irregular triangle begins:
%e A277513         k=1      2      3      4       5      6      7
%e A277513 n=3       4
%e A277513 n=4      12
%e A277513 n=5      24     16
%e A277513 n=6      40     80
%e A277513 n=7      60    240     64
%e A277513 n=8      84    560    448
%e A277513 n=9     112   1120   1792    256
%e A277513 n=10    144   2016   5376   2304
%e A277513 n=11    180   3360  13440  11520    1024
%e A277513 n=12    220   5280  29568  42240   11264
%e A277513 n=13    264   7920  59136 126720   67584   4096
%e A277513 n=14    312  11440 109824 329472  292864  53248
%e A277513 n=15    364  16016 192192 768768 1025024 372736  16384
%e A277513 These are the odd columns with the 1st column removed in the table in A013609.
%t A277513 a = {}; Do[Do[ct = 2^(2k)*Binomial[n - 1, 2k]; AppendTo[a, ct], {k, 1, Floor[(n-1)/2]}], {n, 3, 15}]; a
%Y A277513 Cf. A013609, A188440, A134021, A134022, A134023.
%K A277513 base,nonn,tabf
%O A277513 3,1
%A A277513 _Lei Zhou_, Oct 18 2016