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Sequence is {a(0,n)}, where a(m,0)=1, a(m,n) = a(m-1,n)+a(m,n-1) and a(0,n+1) is such that a(n+1,n+1) = a(0,n).
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1, 0, -3, -4, 7, 11, -62, -14, 581, -1235, -4175, 32520, -48359, -443625, 3136662, -4834644, -60319241, 506792496, -1210299173, -10327456109, 122982395262, -496826354929, -1709350378156, 39417717346686, -259877263864213, 162788318972691, 14331409095176010
EXAMPLE
a(0,n): 1,0,-3,-4,7,...
a(1,n): 1,1,-2,-6,1,...
a(2,n): 1,2,0,-6,-5,...
a(3,n): 1,3,3,-3,-8,...
a(4,n): 1,4,7,4,-4,...
Main diagonal is 1,1,0,-3,-4,..., which is 1 followed by sequence a(0,n).
MAPLE
A111518T := proc(nmax) local a, m, n; a := array(0..nmax, 0..nmax) ; for m from 0 to nmax do a[m, 0] := 1 ; od ; for n from 1 to nmax do a[n, n] := a[0, n-1] ; for m from n+1 to nmax do a[m, n] := a[m-1, n]+a[m, n-1] ; od ; for m from n-1 to 0 by -1 do a[m, n] := a[m+1, n]-a[m+1, n-1] ; od ; od ; RETURN(a) ; end: nmax := 50 ; a := A111518T(nmax) ; r := 0 ; for n from 0 to nmax do printf("%d, ", a[r, n]) ; od; # R. J. Mathar, Sep 26 2006
MATHEMATICA
nmax = 26;
a[_, 0] = 1;
a[m_ /; m > 0, n_ /; n > 0] := a[m, n] = a[m - 1, n] + a[m, n - 1];
sol = Solve[Table[a[n + 1, n + 1] == a[0, n], {n, 0, nmax}], Table[a[0, n], {n, 1, nmax+1}], Integers] // First;
Do[a[m, n] = a[m, n] /. sol, {m, 0, nmax}, {n, 0, nmax}];
Sequence is {a(2,n)}, where a(m,n) is defined at sequence A111518.
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5
1, 2, 0, -6, -5, 7, -43, -107, 410, -308, -5201, 22426, 1694, -462663, 2209642, 47303, -62434277, 381876639, -384111618, -11477555984, 100411394912, -284526009121, -2378813791310, 34944615773187, -187609218526529, -247374733853554, 14024268845995431
EXAMPLE
a(0,n): 1,0,-3,-4,7,...
a(1,n): 1,1,-2,-6,1,...
a(2,n): 1,2,0,-6,-5,...
a(3,n): 1,3,3,-3,-8,...
a(4,n): 1,4,7,4,-4,...
Main diagonal is 1,1,0,-3,-4,..., which is 1 followed by sequence a(0,n).
MAPLE
A111520T := proc(nmax) local a, m, n; a := array(0..nmax, 0..nmax) ; for m from 0 to nmax do a[m, 0] := 1 ; od ; for n from 1 to nmax do a[n, n] := a[0, n-1] ; for m from n+1 to nmax do a[m, n] := a[m-1, n]+a[m, n-1] ; od ; for m from n-1 to 0 by -1 do a[m, n] := a[m+1, n]-a[m+1, n-1] ; od ; od ; RETURN(a) ; end: nmax := 50 ; a := A111520T(nmax) ; r := 2 ; for n from 0 to nmax do printf("%d, ", a[r, n]) ; od; # R. J. Mathar, Sep 26 2006
MATHEMATICA
nmax = 26;
a[_, 0] = 1;
a[m_ /; m > 0, n_ /; n > 0] := a[m, n] = a[m - 1, n] + a[m, n - 1];
sol = Solve[Table[a[n + 1, n + 1] == a[0, n], {n, 0, nmax}], Table[a[0, n], {n, 1, nmax + 1}], Integers] // First;
Do[a[m, n] = a[m, n] /. sol, {m, 0, nmax}, {n, 0, nmax}];
Sequence is {a(3,n)}, where a(m,n) is defined at sequence A111518.
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1, 3, 3, -3, -8, -1, -44, -151, 259, -49, -5250, 17176, 18870, -443793, 1765849, 1813152, -60621125, 321255514, -62856104, -11540412088, 88870982824, -195655026297, -2574468817607, 32370146955580, -155239071570949, -402613805424503, 13621655040570928
EXAMPLE
a(0,n): 1,0,-3,-4,7,...
a(1,n): 1,1,-2,-6,1,...
a(2,n): 1,2,0,-6,-5,...
a(3,n): 1,3,3,-3,-8,...
a(4,n): 1,4,7,4,-4,...
Main diagonal is 1,1,0,-3,-4,..., which is 1 followed by sequence a(0,n).
MAPLE
A111521T := proc(nmax) local a, m, n; a := array(0..nmax, 0..nmax) ; for m from 0 to nmax do a[m, 0] := 1 ; od ; for n from 1 to nmax do a[n, n] := a[0, n-1] ; for m from n+1 to nmax do a[m, n] := a[m-1, n]+a[m, n-1] ; od ; for m from n-1 to 0 by -1 do a[m, n] := a[m+1, n]-a[m+1, n-1] ; od ; od ; RETURN(a) ; end: nmax := 50 ; a := A111521T(nmax) ; m := 3 ; for n from 0 to nmax do printf("%d, ", a[m, n]) ; od; # R. J. Mathar, Sep 26 2006
MATHEMATICA
nmax = 26;
a[_, 0] = 1;
a[m_ /; m > 0, n_ /; n > 0] := a[m, n] = a[m - 1, n] + a[m, n - 1];
sol = Solve[Table[a[n + 1, n + 1] == a[0, n], {n, 0, nmax}], Table[a[0, n], {n, 1, nmax + 1}], Integers] // First;
Do[a[m, n] = a[m, n] /. sol, {m, 0, nmax}, {n, 0, nmax}];
Sequence is {a(4,n)}, where a(m,n) is defined at sequence A111518.
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5
1, 4, 7, 4, -4, -5, -49, -200, 59, 10, -5240, 11936, 30806, -412987, 1352862, 3166014, -57455111, 263800403, 200944299, -11339467789, 77531515035, -118123511262, -2692592328869, 29677554626711, -125561516944238, -528175322368741, 13093479718202187
EXAMPLE
a(0,n): 1,0,-3,-4,7,...
a(1,n): 1,1,-2,-6,1,...
a(2,n): 1,2,0,-6,-5,...
a(3,n): 1,3,3,-3,-8,...
a(4,n): 1,4,7,4,-4,...
Main diagonal is 1,1,0,-3,-4,..., which is 1 followed by sequence a(0,n).
MAPLE
A111522T := proc(nmax) local a, m, n; a := array(0..nmax, 0..nmax) ; for m from 0 to nmax do a[m, 0] := 1 ; od ; for n from 1 to nmax do a[n, n] := a[0, n-1] ; for m from n+1 to nmax do a[m, n] := a[m-1, n]+a[m, n-1] ; od ; for m from n-1 to 0 by -1 do a[m, n] := a[m+1, n]-a[m+1, n-1] ; od ; od ; RETURN(a) ; end: nmax := 50 ; a := A111522T(nmax) ; m := 4 ; for n from 0 to nmax do printf("%d, ", a[m, n]) ; od; # R. J. Mathar, Sep 26 2006
MATHEMATICA
nmax = 26;
a[_, 0] = 1;
a[m_ /; m > 0, n_ /; n > 0] := a[m, n] = a[m - 1, n] + a[m, n - 1];
sol = Solve[Table[a[n + 1, n + 1] == a[0, n], {n, 0, nmax}], Table[a[0, n], {n, 1, nmax + 1}], Integers] // First;
Do[a[m, n] = a[m, n] /. sol, {m, 0, nmax}, {n, 0, nmax}];
Sequence is {a(5,n)}, where a(m,n) is defined at sequence A111518.
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5
1, 5, 12, 16, 12, 7, -42, -242, -183, -173, -5413, 6523, 37329, -375658, 977204, 4143218, -53311893, 210488510, 411432809, -10928034980, 66603480055, -51520031207, -2744112360076, 26933442266635, -98628074677603, -626803397046344, 12466676321155843, -88760048121704842
EXAMPLE
a(0,n): 1,0,-3,-4,7,...
a(1,n): 1,1,-2,-6,1,...
a(2,n): 1,2,0,-6,-5,...
a(3,n): 1,3,3,-3,-8,...
a(4,n): 1,4,7,4,-4,...
Main diagonal is 1,1,0,-3,-4,..., which is 1 followed by sequence a(0,n).
MAPLE
A111523T := proc(nmax) local a, m, n; a := array(0..nmax, 0..nmax) ; for m from 0 to nmax do a[m, 0] := 1 ; od ; for n from 1 to nmax do a[n, n] := a[0, n-1] ; for m from n+1 to nmax do a[m, n] := a[m-1, n]+a[m, n-1] ; od ; for m from n-1 to 0 by -1 do a[m, n] := a[m+1, n]-a[m+1, n-1] ; od ; od ; RETURN(a) ; end: nmax := 50 ; a := A111523T(nmax) ; m := 5 ; for n from 0 to nmax do printf("%d, ", a[m, n]) ; od; # R. J. Mathar, Sep 26 2006
MATHEMATICA
nmax = 27;
a[_, 0] = 1;
a[m_ /; m > 0, n_ /; n > 0] := a[m, n] = a[m - 1, n] + a[m, n - 1];
sol = Solve[Table[a[n + 1, n + 1] == a[0, n], {n, 0, nmax}], Table[a[0, n], {n, 1, nmax + 1}], Integers] // First;
Do[a[m, n] = a[m, n] /. sol, {m, 0, nmax}, {n, 0, nmax}];
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