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 A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k. 9
 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, 64, 1, 34, 81, 108, 120, 125, 127, 128, 1, 55, 149, 208, 236, 248, 253, 255, 256, 1, 89, 274, 401, 464, 492, 504, 509, 511, 512, 1, 144, 504, 773, 912, 976, 1004, 1016, 1021, 1023, 1024 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 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 REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155. LINKS Alois P. Heinz, Rows n = 1..141 FORMULA G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan] 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 T(n,k) = A181695(n,k) - A181695(n-1,k). - Max Alekseyev, Nov 18 2010 Conjecture: Sum_{k=1..n} T(n,k) = A039671(n), n>0. - L. Edson Jeffery, Nov 29 2013 EXAMPLE Triangle begins:   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, 64; Could also be extended to a square array:   1  1  1  1  1  1  1 ...   1  2  2  2  2  2  2 ...   1  3  4  4  4  4  4 ...   1  5  7  8  8  8  8 ...   1  8 13 15 16 16 16 ...   1 13 24 29 31 32 32 ...   1 21 44 56 61 63 64 ... which when read by antidiagonals (downwards) gives A048887. MAPLE 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 # second Maple program: T:= proc(n, k) option remember;       if n=0 or k=1 then 1     else add(T(n-j, k), j=1..min(n, k))       fi     end: seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Oct 23 2011 MATHEMATICA 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 *) CROSSREFS Rows are partial sums of rows of A048004. Cf. A048887, A092921 for other versions. 2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3. Cf. A082267. Sequence in context: A078753 A119443 A209413 * A055888 A094442 A060642 Adjacent sequences:  A126195 A126196 A126197 * A126199 A126200 A126201 KEYWORD nonn,tabl,nice AUTHOR N. J. A. Sloane, Mar 09 2007 EXTENSIONS More terms from R. J. Mathar, Mar 09 2007 STATUS approved

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)