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Search: id:A008289
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| A008289 |
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Triangle read by rows: Q(n,m) = number of partitions of n into m distinct parts, n>=1, m>=1. |
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14
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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1,8
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COMMENT
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Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of terms increases by one at each triangular number).
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.
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LINKS
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T. D. Noe, Rows n=1..200 of triangle, flattened
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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G.f.: Prod_{n>0} (1+y*x^n) = 1 + Sum_{n>0} Q(n, k) y^k x^n.
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
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EXAMPLE
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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:
1;
1;
1,1;
1,1;
1,2;
1,2,1;
1,3,1;
1,3,2;
1,4,3;
1,4,4,1; etc.
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MAPLE
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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
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PROGRAM
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(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))
(PARI) {Q(n, k)=if(n<k|k<1, 0, if(n==1, 1, Q(n-k, k)+Q(n-k, k-1)))} (Hanna)
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CROSSREFS
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Cf. A030699, A104382. Sum of n-th row is A000009. Sum(Q(n, k), k>=1)=A015723(n).
A060016 is another version.
Sequence in context: A084610 A129479 A075104 this_sequence A116679 A146290 A135539
Adjacent sequences: A008286 A008287 A008288 this_sequence A008290 A008291 A008292
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KEYWORD
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nonn,tabf,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Michael Somos, Dec 04 2002
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Nov 20 2006
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