A278295 Number of n X n X n triangular 0..1 arrays with horizontal row sums nondecreasing from top to bottom.

1, 2, 7, 44, 507, 10868, 437908, 33421356, 4860115569, 1353020399536, 723897398723818, 746732196670027756, 1489203154941738419275, 5755222920272113115716592, 43188989125323730167491656884, 630465046596547626339663980200440

Offset

0,2

Comments

With increasing sums we get A003422(n+1). - Alois P. Heinz, Dec 02 2016

With nondecreasing row elements we get A000108(n+1). - Alois P. Heinz, Dec 04 2016

Links

Alois P. Heinz, Table of n, a(n) for n = 0..82

Example

Some solutions for n=3:
    0      0      0      0      0      0      0      0      0
   0 0    0 0    0 0    0 0    0 0    0 0    0 0    0 0    1 0
  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

Maple

noNSumR := proc(n, s)
    binomial(n, s) ;
end proc:
A278295 := proc(n)
    local a, mtot, p, pa, weakp, c, i ;
    a := 0 ;
    mtot := n*(n+1)/2 ;
    for p from 0 to mtot do
        for pa in combinat[partition](p+n) do
            if nops(pa) = n then
                weakp := [seq(op(i, pa)-1, i=1..nops(pa))] ;
                c := 1 ;
                for i from 1 to nops(weakp) do
                    c := c*noNSumR(i, op(i, weakp)) ;
                end do:
                a := a+c ;
            end if;
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Dec 02 2016
# second Maple program:
b:= proc(n, i, k) option remember; `if`(i>n, 1,
      add(binomial(i, j)*b(n, i+1, j), j=k..i))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2016

Mathematica

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 *)

Prog

(PARI) rowsum(rowarr) = sum(x=1, #rowarr, rowarr[x])
is_validcombination(toprow, bottomrow) = if(rowsum(bottomrow) < rowsum(toprow), return(0), return(1))
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
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))
terms(4) \\ print initial four terms

Crossrefs

Cf. A000108, A003422, A256369.

Sequence in context: A172389 A153522 A355109 * A356613 A107354 A006118

Adjacent sequences: A278292 A278293 A278294 * A278296 A278297 A278298

Keyword

nonn

Author

Felix Fröhlich, Nov 30 2016

Extensions

4 more terms from R. J. Mathar, Dec 02 2016

a(0), a(10)-a(15) from Alois P. Heinz, Dec 02 2016

Status

approved