Search: id:A008284
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%I A008284
%S A008284 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,
%T A008284 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,
%U A008284 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
%N A008284 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).
%C A008284 If k > n/2, T(n,k) = P(n-k) = A000041(n-k). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jan 12 2006
%C A008284 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
%D A008284 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.
%D A008284 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
%D A008284 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 219.
%D A008284 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, 
               p. 493.
%H A008284 Franklin T. Adams-Watters, <a href="b008284.txt">First 100 rows, flattened</
               a>
%H A008284 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A008284 H. Bottomley, <a href="a008284.gif">Illustration of initial terms</a>
%H A008284 D. J. Broadhurst and D. Kreimer, <a href="http://arXiv.org/abs/hep-th/
               0001202">Towards cohomology of renormalization...</a>
%H A008284 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A008284.text">
               First 10 rows and more. </a>
%H A008284 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionP.html">Link to a section of The World of Mathematics.</
               a>
%F A008284 T(n, k)=Sum{T(n-k, i)}, 1<=i<=k for 1<=k<=n-1; T(n, n)=1 for n >= 1.
%F A008284 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).
%F A008284 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
%F A008284 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
%F A008284 G.f.=G(t,x)=-1+1/product(1-tx^j,j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Feb 12 2006
%e A008284 Triangle begins:
%e A008284 1;
%e A008284 1,1;
%e A008284 1,1,1;
%e A008284 1,2,1,1;
%e A008284 1,2,2,1,1;
%e A008284 1,3,3,2,1,1; ...
%e A008284 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.
%p A008284 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
%p A008284 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]
%Y A008284 Cf. A000041 (row sums), A038497, A038498, A039805-A039809, A060016. Read 
               from right to left gives A058398. Partial sums of rows gives A026820.
%Y A008284 Column 3 is A001399.
%Y A008284 First difference triangle of triangle A026820.
%Y A008284 Sequence in context: A137350 A166240 A114087 this_sequence A114088 A037306 
               A007424
%Y A008284 Adjacent sequences: A008281 A008282 A008283 this_sequence A008285 A008286 
               A008287
%K A008284 nonn,tabl,nice,easy
%O A008284 1,8
%A A008284 N. J. A. Sloane (njas(AT)research.att.com).

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