A077229
Number of compositions of n where the largest part is less than or equal to the number of parts.
Original entry on oeis.org
1, 1, 1, 3, 5, 11, 23, 48, 98, 204, 421, 863, 1766, 3606, 7341, 14913, 30233, 61175, 123589, 249344, 502443, 1011366, 2033894, 4086975, 8206833, 16469875, 33035611, 66234372, 132745859, 265961487, 532717894, 1066778687, 2135822457, 4275459730, 8557335141, 17125445575, 34268965676, 68568213419, 137187103849, 274458924246
Offset: 0
a(5)=11 since 5 can be written as 1+1+1+1+1, 1+1+1+2, 1+1+2+1, 1+1+3, 1+2+1+1, 1+2+2, 1+3+1, 2+1+1+1, 2+1+2, 2+2+1, or 3+1+1; but not as 2+3 since then the largest part (3) would be greater than the number of parts (2).
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Table[SeriesCoefficient[1 + Sum[x^k*((1-x^k)/(1-x))^k,{k,1,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 01 2014 *)
A077228
Triangle of compositions with a total that is no more than n into exactly k parts each no more than k.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 4, 10, 5, 1, 1, 4, 17, 15, 6, 1, 1, 4, 23, 35, 21, 7, 1, 1, 4, 26, 66, 56, 28, 8, 1, 1, 4, 27, 106, 126, 84, 36, 9, 1, 1, 4, 27, 150, 247, 210, 120, 45, 10, 1, 1, 4, 27, 190, 432, 462, 330, 165, 55, 11, 1, 1, 4, 27, 221, 687, 918, 792, 495, 220, 66
Offset: 0
Rows start: 1; 1,1; 1,3,1; 1,4,4,1; 1,4,10,5,1; 1,4,17,15,6,1; 1,4,23,35,21,7,1; etc. T(6,3)=17 since compositions with 3 parts each no more than 3 and a total no more than 6 are: 1+1+1, 1+1+2, 1+1+3, 1+2+1, 1+2+2, 1+2+3, 1+3+1, 1+3+2, 2+1+1, 2+1+2, 2+1+3, 2+2+1, 2+2+2, 2+3+1, 3+1+1, 3+1+2 and 3+2+1.
Rows eventually start like
A000312. Central diagonal is
A001700. Right hand side and central diagonal is like right hand side of
A007318. Cf.
A077227.
A221833
Triangle, read by rows, where T(n,k) = [x^n] x^k*(1-x^k)^(k-1) / (1-x)^(k-1) for n>=k>=1.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 3, 3, 1, 0, 0, 2, 6, 4, 1, 0, 0, 1, 10, 10, 5, 1, 0, 0, 0, 12, 20, 15, 6, 1, 0, 0, 0, 12, 35, 35, 21, 7, 1, 0, 0, 0, 10, 52, 70, 56, 28, 8, 1, 0, 0, 0, 6, 68, 126, 126, 84, 36, 9, 1, 0, 0, 0, 3, 80, 205, 252, 210, 120, 45, 10, 1
Offset: 1
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 0, 2, 1;
0, 0, 3, 3, 1;
0, 0, 2, 6, 4, 1;
0, 0, 1, 10, 10, 5, 1;
0, 0, 0, 12, 20, 15, 6, 1;
0, 0, 0, 12, 35, 35, 21, 7, 1;
0, 0, 0, 10, 52, 70, 56, 28, 8, 1;
0, 0, 0, 6, 68, 126, 126, 84, 36, 9, 1;
0, 0, 0, 3, 80, 205, 252, 210, 120, 45, 10, 1;
0, 0, 0, 1, 85, 305, 462, 462, 330, 165, 55, 11, 1;
0, 0, 0, 0, 80, 420, 786, 924, 792, 495, 220, 66, 12, 1; ...
in which column sums equal: A000169(k) = k^(k-1) for k>=1.
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{T(n,k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^(k-1),n-k)}
for(n=1,12,for(k=1,n,print1(T(n,k),", "));print(""))
Showing 1-3 of 3 results.