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%I #42 Aug 18 2018 17:28:27
%S 1,1,2,1,3,4,1,5,7,8,1,8,13,15,16,1,13,24,29,31,32,1,21,44,56,61,63,
%T 64,1,34,81,108,120,125,127,128,1,55,149,208,236,248,253,255,256,1,89,
%U 274,401,464,492,504,509,511,512,1,144,504,773,912,976,1004,1016,1021,1023,1024
%N Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.
%C Also has an interpretation as number of binary vectors of length n-1 in which the length of the longest run of 1's is <= k (see A048004). - _N. J. A. Sloane_, Apr 03 2011
%C Higher Order Fibonacci numbers: A126198(n,k) = Sum_{h=0..k} A048004(n,h); for example, A126198(7,3) = Sum_{h=0..3} A048004(7,h) or A126198(7,3) = 1 + 33 + 47 + 27 = 108, the 7th tetranacci number. A048004 row(7) produces A126198 row(7) list of 1,34,81,108,120,125,127,128 which are 1, the 7th Fibonacci, the 7th tribonacci, ... 7th octanacci numbers. - _Richard Southern_, Aug 04 2017
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.
%H Alois P. Heinz, <a href="/A126198/b126198.txt">Rows n = 1..141</a>
%F G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan]
%F T(n,3) = A008937(n) - A008937(n-3) for n>=3. T(n,4) = A107066(n-1) - A107066(n-5) for n>=5. T(n,5) = A001949(n+4) - A001949(n-1) for n>=5. - _R. J. Mathar_, Mar 09 2007
%F T(n,k) = A181695(n,k) - A181695(n-1,k). - _Max Alekseyev_, Nov 18 2010
%F Conjecture: Sum_{k=1..n} T(n,k) = A039671(n), n>0. - _L. Edson Jeffery_, Nov 29 2013
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 3, 4;
%e 1, 5, 7, 8;
%e 1, 8, 13, 15, 16;
%e 1, 13, 24, 29, 31, 32;
%e 1, 21, 44, 56, 61, 63, 64;
%e Could also be extended to a square array:
%e 1 1 1 1 1 1 1 ...
%e 1 2 2 2 2 2 2 ...
%e 1 3 4 4 4 4 4 ...
%e 1 5 7 8 8 8 8 ...
%e 1 8 13 15 16 16 16 ...
%e 1 13 24 29 31 32 32 ...
%e 1 21 44 56 61 63 64 ...
%e which when read by antidiagonals (downwards) gives A048887.
%p A126198 := proc(n,k) coeftayl( x*(1-x^k)/(1-2*x+x^(k+1)),x=0,n); end: for n from 1 to 11 do for k from 1 to n do printf("%d, ",A126198(n,k)); od; od; # _R. J. Mathar_, Mar 09 2007
%p # second Maple program:
%p T:= proc(n, k) option remember;
%p if n=0 or k=1 then 1
%p else add(T(n-j, k), j=1..min(n, k))
%p fi
%p end:
%p seq(seq(T(n, k), k=1..n), n=1..15); # _Alois P. Heinz_, Oct 23 2011
%t rows = 11; t[n_, k_] := Sum[ (-1)^i*2^(n-i*(k+1))*Binomial[ n-i*k, i], {i, 0, Floor[n/(k+1)]}] - Sum[ (-1)^i*2^((-i)*(k+1)+n-1)*Binomial[ n-i*k-1, i], {i, 0, Floor[(n-1)/(k+1)]}]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* _Jean-François Alcover_, Nov 17 2011, after _Max Alekseyev_ *)
%Y Rows are partial sums of rows of A048004. Cf. A048887, A092921 for other versions.
%Y 2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3.
%Y Cf. A082267.
%K nonn,tabl,nice
%O 1,3
%A _N. J. A. Sloane_, Mar 09 2007
%E More terms from _R. J. Mathar_, Mar 09 2007
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