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A008284
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Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1<=k<=n. Also number of partitions of n into k positive parts (1<=k<=n).
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107
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1, 1, 7, 16, 23, 23
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| If k > n/2, T(n,k) = P(n-k) = A000041(n-k). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 12 2006
A002865(n) = Sum(a(n-k+1,k-1): 1<k<=floor((n+2)/2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2007
Contribution from Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010: (Start)
A000041(n+1)=1+ Sum( Sum(T(r,k) :1<=k<=Min(r,n-r+1) ) : 1<=r<=n)
T(n, n-k) is also the number of partitions of k in which the greatest part is at most n-k. (End)
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions, Addison-Wesley Professional, 2005, pp. 38, 45, 50 [From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010]
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 400.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.
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LINKS
| Franklin T. Adams-Watters, First 100 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. Bottomley, Illustration of initial terms
D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...
W. Lang, First 10 rows and more.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| T(n, k)=Sum{T(n-k, i)}, 1<=i<=k for 1<=k<=n-1; T(n, n)=1 for n >= 1.
Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. for k-th column: x^k/(product(1-x^j, j=1..k)) - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2000
G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^(P_n(y)/n), where P_n(y) = Sum_{d|n} eulerphi(n/d)*y^d. - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 13 2004
G.f.=G(t,x)=-1+1/product(1-tx^j,j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 12 2006
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EXAMPLE
| Triangle begins:
1;
1,1;
1,1,1;
1,2,1,1;
1,2,2,1,1;
1,3,3,2,1,1; ...
T(7,3)=4 because we have [3,3,1], [3,2,2], [3,2,1,1] and [3,1,1,1,1], each having greatest part 3; or [5,1,1], [4,2,1], [3,3,1] and [3,2,2] each having 3 parts.
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MAPLE
| G:=-1+1/product(1-t*x^j, j=1..15): Gser:=simplify(series(G, x=0, 17)): for n from 1 to 14 do P[n]:=coeff(Gser, x^n) od: for n from 1 to 14 do seq(coeff(P[n], t^j), j=1..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 12 2006
with(combstruct):for n from 0 to 18 do seq(count(Partition(n), size=m), m = 1 .. n) od; # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 30 2009]
T := proc(n, k) option remember; if k < 0 or n < 0 then 0 elif k = 0 then if n = 0 then 1 else 0 fi else T(n - 1, k - 1) + T(n - k, k) fi end: seq(print(seq(T(n, k), k=1..n)), n=1..14); # Peter Luschny, Jul 24 2011
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MATHEMATICA
| Column[Table[ IntegerPartitions[n, {k}] // Length, {n, 1, 20}, {k, 1, n}], Center] [From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010]
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CROSSREFS
| Cf. A000041 (row sums), A038497, A038498, A039805-A039809, A060016. Read from right to left gives A058398. Partial sums of rows gives A026820.
Column 3 is A001399.
First difference triangle of triangle A026820.
Sequence in context: A137350 A166240 A114087 * A114088 A037306 A194799
Adjacent sequences: A008281 A008282 A008283 * A008285 A008286 A008287
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KEYWORD
| nonn,tabl,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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