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Search: id:A008284
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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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79
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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; listen)
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OFFSET
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1,8
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COMMENT
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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
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REFERENCES
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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. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.
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LINKS
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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, December 1972 [alternative scanned copy].
H. Bottomley, Illustration of initial terms
D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
W. Lang, First 10 rows and more.
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FORMULA
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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
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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; ...
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
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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
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CROSSREFS
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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 A114087 A048789 this_sequence A114088 A037306 A007424
Adjacent sequences: A008281 A008282 A008283 this_sequence A008285 A008286 A008287
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KEYWORD
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nonn,tabl,nice,easy
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AUTHOR
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njas
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