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EXAMPLE
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1
0, "0", 0, 0, 1
1, 0, 16, 0, 36, "0", 16, 0, 1
346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, "128", 48, 0, 1
748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126, 453888, 122016, 25344, 4824, "512", 96, 0, 1
3993445276, 18743463360, 42506546320, 61907282240, 64917874125, 52087325696, 33176621920, 17181584640, 7352761180, 2628808000, 790912656, 201062080, 43284010, 7873920, 1216000, 154496, 17640, "1280", 160, 0, 1
etc..
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MAPLE
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p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 5 do seq(coeff(f(t, n, 4), t, m)/4!^n, m=0..4*n); od;
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