A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.
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
Offset: 1
Examples
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.
References
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.
Links
- Alois P. Heinz, Rows n = 1..141
Crossrefs
Programs
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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
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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 *)
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
Conjecture: Sum_{k=1..n} T(n,k) = A039671(n), n>0. - L. Edson Jeffery, Nov 29 2013
Extensions
More terms from R. J. Mathar, Mar 09 2007
Comments