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proposed
approved
editing
proposed
A doubling of A062344 that gives a skew triangle of coefficients: t(n,m)=(Binomial[2*n, m]*Binomial[2*(n - m), (n - m)]).
Triangle T(n,k) = binomial(2*n,k) *binomial(2*n-2*k,n-k), read by rows; 0<=k<=n.
1,0,2
Row sums are: s(n) = 1, 4, 20, 106, 580, 3244,,...
{2, 14, 86, 510, 2992, 17522, 102818, 605470, 3579980, 21254064}.
tConjecture for row sums: 2*(n+1)*(2*n+1)*s(n) +(-81*n^2+19*n-8)*s(n,m-1)= +10*(Binomial[51*n^2-77*n, m]+30)*Binomial[s(n-2) -500*(n - m1), *(2*n - m3)]*s(n-3)=0. - _R. J. Mathar_, Sep 13 2013
{1},
1;
{2, 2},;
{6, 8, 6},;
{20, 36, 30, 20},;
{70, 160, 168, 112, 70},;
{252, 700, 900, 720, 420, 252},;
{924, 3024, 4620, 4400, 2970, 1584, 924},;
{3432, 12936, 22932, 25480, 20020, 12012, 6006, 3432},;
{12870, 54912, 110880, 141120, 127400, 87360, 48048, 22880, 12870},;
{48620, 231660, 525096, 753984, 771120, 599760, 371280, 190944, 87516, 48620},';
{184756, 972400, 2445300, 3912480, 4476780, 3907008, 2713200, 1550400, 755820, 335920, 184756};
nonn,unedtabl
approved
editing
_Roger L. Bagula _ and _Gary W. Adamson (rlbagulatftn(AT)yahoo.com), _, Sep 17 2008
A doubling of A062344 that gives a skew triangle of coefficients: t(n,m)=(Binomial[2*n, m]*Binomial[2*(n - m), (n - m)]).
1, 2, 2, 6, 8, 6, 20, 36, 30, 20, 70, 160, 168, 112, 70, 252, 700, 900, 720, 420, 252, 924, 3024, 4620, 4400, 2970, 1584, 924, 3432, 12936, 22932, 25480, 20020, 12012, 6006, 3432, 12870, 54912, 110880, 141120, 127400, 87360, 48048, 22880, 12870, 48620
1,2
Row sums are:
{2, 14, 86, 510, 2992, 17522, 102818, 605470, 3579980, 21254064}.
t(n,m)=(Binomial[2*n, m]*Binomial[2*(n - m), (n - m)]).
{1},
{2, 2},
{6, 8, 6},
{20, 36, 30, 20},
{70, 160, 168, 112, 70},
{252, 700, 900, 720, 420, 252},
{924, 3024, 4620, 4400, 2970, 1584, 924},
{3432, 12936, 22932, 25480, 20020, 12012, 6006, 3432},
{12870, 54912, 110880, 141120, 127400, 87360, 48048, 22880, 12870},
{48620, 231660, 525096, 753984, 771120, 599760, 371280, 190944, 87516, 48620},
{184756, 972400, 2445300, 3912480, 4476780, 3907008, 2713200, 1550400, 755820, 335920, 184756}
t[n_, m_] = (Binomial[2*n, m]*Binomial[2*(n - m), (n - m)]); Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]
Cf. A062344.
nonn,uned
Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Sep 17 2008
approved