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 A072574 Triangle T(n,k) of number of compositions (ordered partitions) of n into exactly k distinct parts, 1<=k<=n. 6
 1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 6, 6, 0, 0, 0, 0, 1, 6, 12, 0, 0, 0, 0, 0, 1, 8, 18, 0, 0, 0, 0, 0, 0, 1, 8, 24, 24, 0, 0, 0, 0, 0, 0, 1, 10, 30, 24, 0, 0, 0, 0, 0, 0, 0, 1, 10, 42, 48, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 48, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 60, 120, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS If terms in the compositions did not need to be distinct then the triangle would have values C(n-1,k-1), essentially A007318 offset. LINKS Joerg Arndt, Table of n, a(n) for n = 1..5050 (rows 1..100, flattened). B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97. FORMULA T(n, k) = T(n-k, k)+k*T(n-k, k-1) [with T(n, 0)=1 if n=0 and 0 otherwise] = A000142(k)*A060016(n, k). G.f.: sum(n>=0, n! * z^n * q^((n^2+n)/2) / prod(k=1..n, 1-q^k ) ), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A032020. [Joerg Arndt, Oct 20 2012] EXAMPLE T(6,2)=4 since 6 can be written as 1+5=2+4=4+2=5+1. Triangle starts (trailing zeros omitted for n>=10): [ 1]  1; [ 2]  1, 0; [ 3]  1, 2, 0; [ 4]  1, 2, 0, 0; [ 5]  1, 4, 0, 0, 0; [ 6]  1, 4, 6, 0, 0, 0; [ 7]  1, 6, 6, 0, 0, 0, 0; [ 8]  1, 6, 12, 0, 0, 0, 0, 0; [ 9]  1, 8, 18, 0, 0, 0, 0, 0, 0; [10]  1, 8, 24, 24, 0, 0, ...; [11]  1, 10, 30, 24, 0, 0, ...; [12]  1, 10, 42, 48, 0, 0, ...; [13]  1, 12, 48, 72, 0, 0, ...; [14]  1, 12, 60, 120, 0, 0, ...; [15]  1, 14, 72, 144, 120, 0, 0, ...; [16]  1, 14, 84, 216, 120, 0, 0, ...; [17]  1, 16, 96, 264, 240, 0, 0, ...; [18]  1, 16, 114, 360, 360, 0, 0, ...; [19]  1, 18, 126, 432, 600, 0, 0, ...; [20]  1, 18, 144, 552, 840, 0, 0, ...; These rows (without the zeros) are shown in the Richmond/Knopfmacher reference. PROG (PARI) N=21;  q='q+O('q^N); gf=sum(n=0, N, n! * z^n * q^((n^2+n)/2) / prod(k=1, n, 1-q^k ) ); /* print triangle: */ gf -= 1; /* remove row zero */ P=Pol(gf, 'q); { for (n=1, N-1,     p = Pol(polcoeff(P, n), 'z);     p += 'z^(n+1);  /* preserve trailing zeros */     v = Vec(polrecip(p));     v = vector(n, k, v[k]); /* trim to size n */     print(v); ); } /* Joerg Arndt, Oct 20 2012 */ CROSSREFS Columns (offset) include A057427 and A052928. Row sums are A032020. Cf. A060016, A072575, A072576, A216652. Sequence in context: A263844 A079644 A072705 * A293595 A261249 A058650 Adjacent sequences:  A072571 A072572 A072573 * A072575 A072576 A072577 KEYWORD nonn,tabl AUTHOR Henry Bottomley, Jun 21 2002 STATUS approved

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Last modified February 21 23:29 EST 2019. Contains 320381 sequences. (Running on oeis4.)