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A recursive triangular sequence with row sums (5^(n - 1)*(n + 3)!)/12: A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + 5 *(2 + n) (13 + 5* n)*A(n - 2, k - 1).
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%I #2 Mar 30 2012 17:34:29

%S 2,25,25,2,1496,2,2,26248,26248,2,2,28910,2042176,28910,2,2,32352,

%T 47217646,47217646,32352,2,2,36674,109695598,4505535452,109695598,

%U 36674,2,2,41976,195465072,129741992950,129741992950,195465072,41976,2,2

%N A recursive triangular sequence with row sums (5^(n - 1)*(n + 3)!)/12: A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + 5 *(2 + n) (13 + 5* n)*A(n - 2, k - 1).

%C Row sums are:(5^(n - 1)*(n + 3)!)/12:

%C {2, 50, 1500, 52500, 2100000, 94500000, 4725000000, 259875000000,

%C 15592500000000, 1013512500000000, 70945875000000000, 5320940625000000000,...}.

%C The whole sequence is adjusted to get the lowest terms.

%F A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + 5 *(2 + n) (13 + 5* n)*A(n - 2, k - 1).

%e {2},

%e {25, 25},

%e {2, 1496, 2},

%e {2, 26248, 26248, 2},

%e {2, 28910, 2042176, 28910, 2},

%e {2, 32352, 47217646, 47217646, 32352, 2},

%e {2, 36674, 109695598, 4505535452, 109695598, 36674, 2},

%e {2, 41976, 195465072, 129741992950, 129741992950, 195465072, 41976, 2},

%e {2, 48358, 312497108, 479866415642, 14632142077780, 479866415642, 312497108, 48358, 2},

%e {2, 55920, 471214746, 1219036884910, 505536741844422, 505536741844422, 1219036884910, 471214746, 55920, 2},

%e {2, 64762, 685013026, 2600745317016, 2627765335866972, 65685141467476444, 2627765335866972, 2600745317016, 685013026, 64762, 2},

%e {2, 74984, 970828988, 5009337682102, 8859644563074088, 2651605657628339836, 2651605657628339836, 8859644563074088, 5009337682102, 970828988, 74984, 2}

%t Clear[A]; A[1, 1] = 2; A[2, 1] := A[2, 2] = 5* 60/12; A[3, 2] = (18000 - 48)/12; A[4, 2] = (630000/2 - 24)/12; A[4, 3] = (630000/2 - 24)/12; A[n_, 1] := 2; A[n_, n_] := 2;

%t A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + 5 *(2 + n) (13 + 5* n)*A[n - 2, k - 1];

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

%t Flatten[a]

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

%t Table[Apply[Plus, a[[n]]]/(5^(n - 1)*(n + 3)!), {n, 1, 12}];

%K nonn,tabl,uned

%O 1,1

%A _Roger L. Bagula_, Jan 01 2009