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A triangular sequence recursion: A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (-12 + 5 n) (-9 + 5 n)*A(n - 2, k - 1).
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%I #2 Mar 30 2012 17:34:29

%S 2,3,3,2,44,2,2,310,310,2,2,728,9772,728,2,2,1486,127680,127680,1486,

%T 2,2,2684,564510,6099016,564510,2684,2,2,4422,1857042,117489766,

%U 117489766,1857042,4422,2,2,6800,5050056,789984688,7480610540,789984688,5050056

%N A triangular sequence recursion: A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (-12 + 5 n) (-9 + 5 n)*A(n - 2, k - 1).

%C Row sums are:2*Product[5*k + 3, {k, 0, n - 1}];

%C {2, 6, 48, 624, 11232, 258336, 7233408, 238702464, 9070693632, 390039826176,

%C 18721911656448, 992261317791744,...}.

%F A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (-12 + 5 n) (-9 + 5 n)*A(n - 2, k - 1).

%e {2},

%e {3, 3},

%e {2, 44, 2},

%e {2, 310, 310, 2},

%e {2, 728, 9772, 728, 2},

%e {2, 1486, 127680, 127680, 1486, 2},

%e {2, 2684, 564510, 6099016, 564510, 2684, 2},

%e {2, 4422, 1857042, 117489766, 117489766, 1857042, 4422, 2},

%e {2, 6800, 5050056, 789984688, 7480610540, 789984688, 5050056, 6800, 2},

%e {2, 9918, 11946332, 3688306180, 191319650656, 191319650656, 3688306180, 11946332, 9918, 2},

%e {2, 13876, 25406650, 13689263280, 1757597669700, 15179286949432, 1757597669700, 13689263280, 25406650, 13876, 2},

%e {2, 18774, 49699790, 42959290666, 10800260461620, 485287389425020, 485287389425020, 10800260461620, 42959290666, 49699790, 18774, 2}

%t A[1, 1] = 2; A[2, 1] := A[2, 2] = 3;

%t A[3, 2] = 48 - 4; A[4, 2] = 624/2 - 2; A[4, 3] = 624/2 - 2;

%t A[n_, 1] := 2; A[n_, n_] := 2;

%t A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + (-12 + 5 n) (-9 + 5 n)*A[n - 2, k - 1];

%t a = Table[A[n, k], {n, 12}, {k, n}]; Flatten[a]

%t Table[Apply[Plus, a[[n]]], {n, 1, 12}];

%K nonn,tabl

%O 1,1

%A _Roger L. Bagula_, Jan 03 2009