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A232539 Triangle read by rows: T(n,k) = number of partitions of n into at most four parts in which the largest part is equal to k, 0 <= k <= n.

%I A232539 #26 May 27 2025 10:21:47
%S A232539 1,0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,0,2,2,1,1,0,0,2,3,2,1,1,0,0,1,3,3,2,
%T A232539 1,1,0,0,1,3,4,3,2,1,1,0,0,0,3,4,4,3,2,1,1,0,0,0,2,5,5,4,3,2,1,1,0,0,
%U A232539 0,1,4,6,5,4,3,2,1,1,0,0,0,1,4,6,7,5,4,3,2,1,1
%N A232539 Triangle read by rows: T(n,k) = number of partitions of n into at most four parts in which the largest part is equal to k, 0 <= k <= n.
%C A232539 Also number of partitions of n into k parts with parts in the range 1..4.
%H A232539 Louis Comtet, <a href="https://doi.org/10.1007/978-94-010-2196-8">Advanced Combinatorics</a>, Reidel (1974).
%F A232539 G.f.: 1/((1-u*t)*(1-u*t^2)*(1-u*t^3)*(1-u*t^4)). - [Comtet p. 97 [2c]]. - _R. J. Mathar_, May 27 2025
%e A232539 Triangle T{n,k} begins:
%e A232539   1;
%e A232539   0, 1;
%e A232539   0, 1, 1;
%e A232539   0, 1, 1, 1;
%e A232539   0, 1, 2, 1, 1;
%e A232539   0, 0, 2, 2, 1, 1;
%e A232539   0, 0, 2, 3, 2, 1, 1;
%e A232539   0, 0, 1, 3, 3, 2, 1, 1;
%e A232539   0, 0, 1, 3, 4, 3, 2, 1, 1;
%e A232539   0, 0, 0, 3, 4, 4, 3, 2, 1, 1;
%e A232539   ...
%p A232539 maxp := 4 :
%p A232539 gf := 1/mul(1-u*t^i,i=1..maxp) :
%p A232539 for n from 0 to 13 do
%p A232539     for m from 0 to n do
%p A232539         coeftayl(gf,t=0,n) ;
%p A232539         coeftayl(%,u=0,m) ;
%p A232539         printf("%d ",%);
%p A232539     end do:
%p A232539     printf("\n") ;
%p A232539 end do: # _R. J. Mathar_, May 27 2025
%Y A232539 Cf. A001400 (row sums), A219237, A233292 (row partial sums), A145362 (parts <=2), A339884 (parts <=3).
%K A232539 nonn,tabl
%O A232539 0,13
%A A232539 _L. Edson Jeffery_, Jan 02 2014