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4495292802000,      2327932807200,       667458792000,        81729648000;\n  8:   39531606447181,   276571337613690,    844291094921910,   1468765934672400,   1592486639944200,   1101873471098400,    475106703271200,    116709937344000,    12504636144000;\n  9: 8661323866026007, 68190617712846006, 238044766594758480, 483572419758991080, 629951395986549000, 545714976479072400, 314351037648648000, 116101378384992000, 24946749107280000, 2375880867360000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+6*x)*P(n-1,x) + (12+21*x+9*x^2)*d/dx P(n-1,x) + (3+8*x+7*x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(7 + 6*x)*P[n - 1, x] + (12 + 21*x + 9*x^2)*D[P[n - 1, x], x] + (3 + 8*x + 7*x^2 + 2*x^3)*D[P[n - 1, 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] = (7 + 6*x)*v[i - 1] + (12 + 21*x + 9*x^2)*deriv(v[i - 1], x) + (3 + 8*x + 7*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x)); 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-4ef34abfaa61f0d6","steps":["x \u21a6 x+1"],"edges":["e-0ab5bdecdd185a22"],"text":"A241171 \u2192 x \u21a6 x+1"},"headline":"A241171 \u2192 x \u21a6 x+1","chain":null}