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A229896 Sizes of logical groups of the same integer in A229895.

%I A229896 #18 Aug 16 2024 17:01:11
%S A229896 1,1,4,1,5,27,1,7,37,256,1,9,61,369,3125,1,11,91,671,4651,46656,1,13,
%T A229896 127,1105,9031,70993,823543,1,15,169,1695,15961,144495,1273609,
%U A229896 16777216,1,17,217,2465,26281,269297,2685817,26269505,387420489,1,19,271,3439
%N A229896 Sizes of logical groups of the same integer in A229895.
%C A229896 The two ones at the start of the parent sequence represent parent and child 1-tuples in the grandparent sequence [(1) and (2) respectively], hence this sequence also starts with 1, 1 rather than 2, which would otherwise be a more sensible way to describe the pair of ones.
%C A229896 All other elements are effectively run-lengths of strings of the same integer in A229895.
%C A229896 The first occurrence of an integer, n, in the parent sequence, is the first of a run of n^n elements of value n. For later occurrences, the run length is n^k-(n-1)^k where k is the size of the k-tuple in the grandparent sequence, A229873.
%C A229896 The elements can be arranged into a triangle thus:
%C A229896 .... 1
%C A229896 .... 1, 4
%C A229896 .... 1, 5, 27
%C A229896 .... 1, 7, 37, 256
%C A229896 .... 1, 9, 61, 369, 3125
%C A229896 .... etc.
%C A229896 where the n-th line is:
%C A229896 .... n^1-(n-1)^1, n^2-(n-1)^2, ..., n^(k-1)-(n-1)^(k-1), n^n; 1 <= k < n
%C A229896 The first terms, for sufficiently large n simplifying as:
%C A229896 .... 1, 2n-1, 3n^2-3n+1, 4n^3-6n^2+4n-1, etc.
%C A229896 Row sums are first differences of A031972, and thus the cumulative sum of rows at the end of each row is A031972 itself, i.e., n*(n^n - 1)/(n-1).
%H A229896 Carl R. White, <a href="/A229896/b229896.txt">Rows 1..44 for triangle, flattened</a>
%p A229896 T := proc (n, k) if k < n then n^k-(n-1)^k elif k = n then n^n else end if end proc: for n to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form - _Emeric Deutsch_, Jan 30 2017
%o A229896 (bc) /* GNU bc */ for(n=1;n<=10;n++)for(p=1;p<=n;p++){if(p==n){t=n^n}else{t=n^p-(n-1)^p};print t,","};print "...\n"
%Y A229896 Cf. A229873, A229895, A031972.
%K A229896 nonn,easy,tabl
%O A229896 1,3
%A A229896 _Carl R. White_, Oct 03 2013