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A292835 Triangle C(nu,m) read by rows: the number of n-color odd compositions of nu into m parts.

%I A292835 #15 Aug 15 2023 10:53:26
%S A292835 1,0,1,0,0,1,0,3,0,1,0,0,6,0,1,0,5,0,9,0,1,0,0,19,0,12,0,1,0,7,0,42,0,
%T A292835 15,0,1,0,0,44,0,74,0,18,0,1,0,9,0,138,0,115,0,21,0,1,0,0,85,0,316,0,
%U A292835 165,0,24,0,1,0,11,0,363,0,605,0,224,0,27,0,1,0,0,146,0,1059,0,1032,0,292,0,30,0,1
%N A292835 Triangle C(nu,m) read by rows: the number of n-color odd compositions of nu into m parts.
%H A292835 Y-h. Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Guo/guo4.html">Some n-Color Compositions</a>, J. Int. Seq. 15 (2012) 12.1.2, eq. (7).
%H A292835 Y.-h. Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Guo/guo9.html">n-Color Odd Self-Inverse Compositions</a>, J. Int. Seq. 17 (2014) # 14.10.5, eq (1).
%e A292835 The triangle starts in row nu=0 with columns 0<=m<=nu as
%e A292835 1;
%e A292835 0,1;
%e A292835 0,0,1;
%e A292835 0,3,0,1;
%e A292835 0,0,6,0,1;
%e A292835 0,5,0,9,0,1;
%e A292835 0,0,19,0,12,0,1;
%e A292835 0,7,0,42,0,15,0,1;
%e A292835 0,0,44,0,74,0,18,0,1;
%e A292835 0,9,0,138,0,115,0,21,0,1;
%e A292835 0,0,85,0,316,0,165,0,24,0,1;
%e A292835 0,11,0,363,0,605,0,224,0,27,0,1;
%p A292835 A292835 := proc(nu,m)
%p A292835     local a,j,i ;
%p A292835     a := 0 ;
%p A292835     for j from 0 to m do
%p A292835         i := (nu-m)/2-j ;
%p A292835         if type(i,integer) then
%p A292835             a := a+binomial(2*m+i-1,2*m-1)*binomial(m,j) ;
%p A292835         end if;
%p A292835     end do:
%p A292835     a;
%p A292835 end proc:
%p A292835 seq(seq(A292835(nu,m),m=0..nu),nu=0..12) ;
%t A292835 A292835[nu_, m_] := Module[{a, j, i}, a = 0; For[j = 0, j <= m, j++, i = (nu - m)/2 - j; If[IntegerQ[i], a = a + Binomial[2m + i - 1, 2m - 1]* Binomial[m, j]]]; a];
%t A292835 Table[Table[A292835[nu, m], {m, 0, nu}], {nu, 0, 12}] // Flatten (* _Jean-François Alcover_, Aug 15 2023, after Maple code *)
%Y A292835 Cf. A119749 (row sums), A005900 (column m=3), A300437.
%K A292835 nonn,tabl,easy
%O A292835 0,8
%A A292835 _R. J. Mathar_, Sep 24 2017