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 A278295 Number of n X n X n triangular 0..1 arrays with horizontal row sums nondecreasing from top to bottom. 1
 1, 2, 7, 44, 507, 10868, 437908, 33421356, 4860115569, 1353020399536, 723897398723818, 746732196670027756, 1489203154941738419275, 5755222920272113115716592, 43188989125323730167491656884, 630465046596547626339663980200440 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 19 07:05 EDT 2024. Contains 372666 sequences. (Running on oeis4.)