A278295 - OEIS

%I #54 Feb 28 2017 22:55:21

%S 1,2,7,44,507,10868,437908,33421356,4860115569,1353020399536,

%T 723897398723818,746732196670027756,1489203154941738419275,

%U 5755222920272113115716592,43188989125323730167491656884,630465046596547626339663980200440

%N Number of n X n X n triangular 0..1 arrays with horizontal row sums nondecreasing from top to bottom.

%C With increasing sums we get A003422(n+1). - _Alois P. Heinz_, Dec 02 2016

%C With nondecreasing row elements we get A000108(n+1). - _Alois P. Heinz_, Dec 04 2016

%H Alois P. Heinz, <a href="/A278295/b278295.txt">Table of n, a(n) for n = 0..82</a>

%e Some solutions for n=3:

%e     0      0      0      0      0      0      0      0      0

%e    0 0    0 0    0 0    0 0    0 0    0 0    0 0    0 0    1 0

%e   0 0 0  1 0 0  0 1 0  0 0 1  1 1 0  1 0 1  0 1 1  1 1 1  1 0 0

%p noNSumR := proc(n,s)

%p     binomial(n,s) ;

%p end proc:

%p A278295 := proc(n)

%p     local a,mtot,p,pa,weakp,c,i ;

%p     a := 0 ;

%p     mtot := n*(n+1)/2 ;

%p     for p from 0 to mtot do

%p         for pa in combinat[partition](p+n) do

%p             if nops(pa) = n then

%p                 weakp := [seq(op(i,pa)-1,i=1..nops(pa))] ;

%p                 c := 1 ;

%p                 for i from 1 to nops(weakp) do

%p                     c := c*noNSumR(i,op(i,weakp)) ;

%p                 end do:

%p                 a := a+c ;

%p             end if;

%p         end do:

%p     end do:

%p     a;

%p end proc: # _R. J. Mathar_, Dec 02 2016

%p # second Maple program:

%p b:= proc(n, i, k) option remember; `if`(i>n, 1,

%p       add(binomial(i, j)*b(n, i+1, j), j=k..i))

%p     end:

%p a:= n-> b(n, 0$2):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 02 2016

%t b[n_, i_, k_] := b[n, i, k] = If[i>n, 1, Sum[Binomial[i, j]*b[n, i+1, j], {j, k, i}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 28 2017, after _Alois P. Heinz_ *)

%o (PARI) rowsum(rowarr) = sum(x=1, #rowarr, rowarr[x])

%o is_validcombination(toprow, bottomrow) = if(rowsum(bottomrow) < rowsum(toprow), return(0), return(1))

%o nextrowcomb(rowarr) = my(k=#rowarr, i=0); while(rowarr[k]==1, rowarr[k]=0; i++; k--); while(rowarr[k]==0 && k > 1, k--); if(rowarr[k]==1, rowarr[k]=0; rowarr[k+1]=1; k=k+2; while(i > 0, rowarr[k]=1; k++; i--), for(x=k, k+i, rowarr[x]=1)); rowarr

%o terms(n) = my(toprows=[[0], [1]], bottomrow=[0, 0], validrows=[]); while(1, for(k=1, #toprows, if(is_validcombination(toprows[k], bottomrow), validrows=concat(validrows, [bottomrow]))); if(vecmin(bottomrow)==1, bottomrow=vector(#bottomrow+1); print1(#validrows, ", "); toprows=validrows; validrows=[], bottomrow=nextrowcomb(bottomrow)); if(#bottomrow==n+2, break))

%o terms(4) \\ print initial four terms

%Y Cf. A000108, A003422, A256369.

%K nonn

%O 0,2

%A _Felix Fröhlich_, Nov 30 2016

%E 4 more terms from _R. J. Mathar_, Dec 02 2016

%E a(0), a(10)-a(15) from _Alois P. Heinz_, Dec 02 2016