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972,         648;\n  4:    81,     2430,      12150,       19440,         9720;\n  5:   243,    15066,     131220,      379080,       437400,       174960;\n  6:   729,    91854,    1316574,     6123600,     12247200,     11022480,      3674160;\n  7:  2187,   555498,   12675852,    89282088,    275562000,    418854240,    308629440,     88179840;\n  8:  6561,  3346110,  119082150,  1223495280,   5472661320,  12499492320,  15277157280,   9523422720,   2380855680;\n  9: 19683, 20116026, 1101854340, 16110929160, 100442349000, 323498765520, 583309641600, 595213920000, 321415516800, 71425670400;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-dfedc6a36c97eba4","steps":["x \u21a6 x/(x+1)"],"edges":["e-a57608fc217f45d7"],"text":"A257620 \u2192 x \u21a6 x/(x+1)"},"headline":"A257620 \u2192 x \u21a6 x/(x+1)","chain":null}