A168396 Triangle, T(n,k) = number of compositions a(1),...,a(j) of n with a(1) = k, such that a(i+1) <= a(i) + 1 for 1 <= i < j.
1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 9, 6, 4, 2, 1, 1, 15, 11, 7, 4, 2, 1, 1, 26, 19, 12, 7, 4, 2, 1, 1, 45, 33, 21, 13, 7, 4, 2, 1, 1, 78, 57, 37, 22, 13, 7, 4, 2, 1, 1, 135, 99, 64, 39, 23, 13, 7, 4, 2, 1, 1, 234, 172, 112, 68, 40, 23, 13, 7, 4, 2, 1, 1, 406, 298, 194, 119, 70, 41, 23, 13, 7, 4, 2, 1, 1
Offset: 1
Examples
First 16 rows of triangle: . 1: 1 . 2: 1 1 . 3: 2 1 1 . 4: 3 2 1 1 . 5: 5 4 2 1 1 . 6: 9 6 4 2 1 1 . 7: 15 11 7 4 2 1 1 . 8: 26 19 12 7 4 2 1 1 . 9: 45 33 21 13 7 4 2 1 1 . 10: 78 57 37 22 13 7 4 2 1 1 . 11: 135 99 64 39 23 13 7 4 2 1 1 . 12: 234 172 112 68 40 23 13 7 4 2 1 1 . 13: 406 298 194 119 70 41 23 13 7 4 2 1 1 . 14: 704 518 337 207 123 71 41 23 13 7 4 2 1 1 . 15: 1222 898 586 360 214 125 72 41 23 13 7 4 2 1 1 . 16: 2120 1559 1017 626 373 218 126 72 41 23 13 7 4 2 1 1
Links
- Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
Programs
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Haskell
a168396 n k = a168396_tabl !! (n-1) !! (k-1) a168396_row n = a168396_tabl !! (n-1) a168396_tabl = [1] : f [[1]] where f xss = ys : f (ys : xss) where ys = (map sum $ zipWith take [2..] xss) ++ [1] -- Reinhard Zumkeller, Sep 13 2013 -
Maple
b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, j+1), j=1..min(n, k))) end: T:= (n, k)-> b(n-k, k+1): seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Sep 19 2013 -
Mathematica
t[n_, k_] /; k > n = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[ t[n-k, j], {j, 1, k+1}]; Flatten[ Table[ t[n, k], {n, 1, 13}, {k, 1, n}] ](* Jean-François Alcover, Feb 17 2012, after Pari *) -
PARI
T(n,k)=if(k>=n,k==n,sum(j=1,k+1,T(n-k,j)))
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PARI
Tm(n)=local(m);m=matrix(n,n);for(i=1,n,for(j=1,i,m[i,j]=if(i==j,1,sum(k=1,j+1,m[i-j,k]))));m
Comments