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%I #23 Aug 30 2018 20:09:52
%S 1,2,0,3,1,0,5,2,1,0,7,6,2,1,0,11,12,6,2,1,0,15,26,14,6,2,1,0,22,50,
%T 33,14,6,2,1,0,30,97,72,34,14,6,2,1,0,42,180,156,77,34,14,6,2,1,0,56,
%U 332,328,173,78,34,14,6,2,1,0,77,600,681,378,177,78,34,14,6,2,1,0,101,1078,1393,818,393,178,78,34,14,6,2,1,0
%N Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step k; n>=1, 0<=k<n.
%C T(n,k) is the number of compositions [p(1), p(2), ..., p(k)] of n such that max(p(j) - p(j-1)) == k.
%C The first column is A000041 (partition numbers).
%C Sum of first and second column is A003116.
%C Sum of the first three columns is A224959.
%C The second columns deviates from A054454 after the term 600.
%C Row sums are A011782.
%H Joerg Arndt and Alois P. Heinz, <a href="/A225084/b225084.txt">Rows n = 1..141, flattened</a>
%e Triangle starts:
%e 01: 1,
%e 02: 2, 0,
%e 03: 3, 1, 0,
%e 04: 5, 2, 1, 0,
%e 05: 7, 6, 2, 1, 0,
%e 06: 11, 12, 6, 2, 1, 0,
%e 07: 15, 26, 14, 6, 2, 1, 0,
%e 08: 22, 50, 33, 14, 6, 2, 1, 0,
%e 09: 30, 97, 72, 34, 14, 6, 2, 1, 0,
%e 10: 42, 180, 156, 77, 34, 14, 6, 2, 1, 0,
%e 11: 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0,
%e 12: 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0,
%e 13: 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0,
%e 14: 135, 1917, 2821, 1746, 863, 397, 178, 78, 34, 14, 6, 2, 1, 0,
%e 15: 176, 3393, 5660, 3695, 1872, 877, 398, 178, 78, 34, 14, 6, 2, 1, 0,
%e ...
%e The fifth row corresponds to the following statistics:
%e #: M composition
%e 01: 0 [ 1 1 1 1 1 ]
%e 02: 1 [ 1 1 1 2 ]
%e 03: 1 [ 1 1 2 1 ]
%e 04: 2 [ 1 1 3 ]
%e 05: 1 [ 1 2 1 1 ]
%e 06: 1 [ 1 2 2 ]
%e 07: 2 [ 1 3 1 ]
%e 08: 3 [ 1 4 ]
%e 09: 0 [ 2 1 1 1 ]
%e 10: 1 [ 2 1 2 ]
%e 11: 0 [ 2 2 1 ]
%e 12: 1 [ 2 3 ]
%e 13: 0 [ 3 1 1 ]
%e 14: 0 [ 3 2 ]
%e 15: 0 [ 4 1 ]
%e 16: 0 [ 5 ]
%e There are 7 compositions with no up-step (M=0), 6 with M=1, 2 with M=2, and 1 with M=3.
%p b:= proc(n, v) option remember; `if`(n=0, 1, add((p->
%p `if`(i<v, add(coeff(p, x, h)*x^`if`(h<v-i, v-i, h),
%p h=0..degree(p)), p))(b(n-i, i)), i=1..n))
%p end:
%p T:= n-> seq(coeff(b(n, 0), x, i), i=0..n-1):
%p seq(T(n), n=1..14); # _Alois P. Heinz_, Feb 22 2014
%t b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[Function[{p}, If[i<v, Sum[Coefficient[p, x, h]*x^If[h<v-i, v-i, h], {h, 0, Exponent[p, x]}], p]][b[n-i, i]], {i, 1, n}]] ; T[n_] := Table[Coefficient[b[n, 0], x, i], {i, 0, n-1}]; Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)
%Y Cf. A225085 (partial sums of rows).
%Y T(2n,n) gives A059570.
%K nonn,tabl
%O 1,2
%A _Joerg Arndt_, Apr 27 2013
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