Search: id:A008289
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%I A008289
%S A008289 1,1,1,1,1,1,1,2,1,2,1,1,3,1,1,3,2,1,4,3,1,4,4,1,1,5,5,1,1,5,7,2,1,6,8,
%T A008289 3,1,6,10,5,1,7,12,6,1,1,7,14,9,1,1,8,16,11,2,1,8,19,15,3,1,9,21,18,5,
%U A008289 1,9,24,23,7,1,10,27,27,10,1,1,10,30,34,13,1,1,11,33,39,18,2,1,11,37
%N A008289 Triangle read by rows: Q(n,m) = number of partitions of n into m distinct 
               parts, n>=1, m>=1.
%C A008289 Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of 
               terms increases by one at each triangular number).
%D A008289 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.
%H A008289 T. D. Noe, <a href="b008289.txt">Rows n=1..200 of triangle, flattened</
               a>
%H A008289 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionQ.html">Link to a section of The World of Mathematics.</
               a>
%F A008289 G.f.: Prod_{n>0} (1+y*x^n) = 1 + Sum_{n>0} Q(n, k) y^k x^n.
%F A008289 Q(n, k) = Q(n-k, k) + Q(n-k, k-1) for n>k>=1, with Q(1, 1)=1, Q(n, 0)=0 
               (n>=1). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 04 2005
%e A008289 Q(8,3)=2 since 8 can be written in 2 ways as sum of 3 distinct positive 
               integers: 5+2+1 and 4+3+1.Triangle starts:
%e A008289 1;
%e A008289 1;
%e A008289 1,1;
%e A008289 1,1;
%e A008289 1,2;
%e A008289 1,2,1;
%e A008289 1,3,1;
%e A008289 1,3,2;
%e A008289 1,4,3;
%e A008289 1,4,4,1; etc.
%p A008289 g:=product(1+t*x^j,j=1..40): gser:=simplify(series(g,x=0,32)): P[0]:=1: 
               for n from 1 to 30 do P[n]:=sort(coeff(gser,x^n)) od: for n from 
               1 to 25 do seq(coeff(P[n],t,j),j=1..floor((sqrt(8*n+1)-1)/2)) od; 
               # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Feb 21 2006
%o A008289 (PARI) Q(n,k)=if(k<0|k>n,0,polcoeff(polcoeff(prod(i=1,n,1+y*x^i,1+x*O(x^n)),
               n),k))
%o A008289 (PARI) {Q(n,k)=if(n<k|k<1,0,if(n==1,1,Q(n-k,k)+Q(n-k,k-1)))} (Hanna)
%Y A008289 Cf. A030699, A104382. Sum of n-th row is A000009. Sum(Q(n, k), k>=1)=A015723(n).
%Y A008289 A060016 is another version.
%Y A008289 Sequence in context: A084610 A129479 A075104 this_sequence A116679 A146290 
               A135539
%Y A008289 Adjacent sequences: A008286 A008287 A008288 this_sequence A008290 A008291 
               A008292
%K A008289 nonn,tabf,easy,nice
%O A008289 1,8
%A A008289 N. J. A. Sloane (njas(AT)research.att.com).
%E A008289 Additional comments from Michael Somos, Dec 04 2002
%E A008289 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Nov 20 2006

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