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b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"87e39f5e0ae6bba0a805d4c5aa659277b4c095f4fd2a6d504654e0e18da36e81","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-25ac1cc25c118491","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,        2;\n  2: 1,       14,          14;\n  3: 1,      106,         278,          174;\n  4: 1,      838,        4678,         7162,          3322;\n  5: 1,     6690,       80660,       229680,        245464,         89756;\n  6: 1,    53502,     1486510,      7063340,      13087160,      10668248,       3211420;\n  7: 1,   427994,    29023194,    220635290,     635669740,     869497448,     572000604,     146132808;\n  8: 1,  3423926,   587659478,   7154906294,   30406730050,   62298056248,   67280173924,   37020644528,    8202467544;\n  9: 1, 27391378, 12131989688, 242901190412, 1479654376772, 4277047855828, 6755155744492, 6010368385792, 2837891772656, 554489060200;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(4*n-8)*x+(3*n^2-21*n+23)*x^2+(9*n^2-39*n+32)*x^3+(9*n^2-31*n+22)*x^4+(3*n^2-9*n+6)*x^5)*P(n-1,x) + (7*x-(6*n-37)*x^2-(24*n-81)*x^3-(36*n-91)*x^4-(24*n-52)*x^5-(6*n-12)*x^6)*d/dx P(n-1,x) + (3*x^2+15*x^3+30*x^4+30*x^5+15*x^6+3*x^7)*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[Together[((1 - (4*n - 8)*x + (3*n^2 - 21*n + 23)*x^2 + (9*n^2 - 39*n + 32)*x^3 + (9*n^2 - 31*n + 22)*x^4 + (3*n^2 - 9*n + 6)*x^5)*P[n - 1, x] + (7*x - (6*n - 37)*x^2 - (24*n - 81)*x^3 - (36*n - 91)*x^4 - (24*n - 52)*x^5 - (6*n - 12)*x^6)*D[P[n - 1, x], x] + (3*x^2 + 15*x^3 + 30*x^4 + 30*x^5 + 15*x^6 + 3*x^7)*D[P[n - 1, x], {x, 2}])/(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] = ((1 - (4*n - 8)*x + (3*n^2 - 21*n + 23)*x^2 + (9*n^2 - 39*n + 32)*x^3 + (9*n^2 - 31*n + 22)*x^4 + (3*n^2 - 9*n + 6)*x^5)*v[i - 1] + (7*x - (6*n - 37)*x^2 - (24*n - 81)*x^3 - (36*n - 91)*x^4 - (24*n - 52)*x^5 - (6*n - 12)*x^6)*deriv(v[i - 1], x) + (3*x^2 + 15*x^3 + 30*x^4 + 30*x^5 + 15*x^6 + 3*x^7)*deriv(deriv(v[i - 1], x), 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-25ac1cc25c118491","steps":["x \u21a6 x/(x+1)"],"edges":["e-a5ddde62de67fc97"],"text":"A166962 \u2192 x \u21a6 x/(x+1)"},"headline":"A166962 \u2192 x \u21a6 x/(x+1)","chain":null}