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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":"3ce3812350525a2e51d7d5f997655243fec4759c146191fce632453aecf70d26","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e18d516553291356","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,2].","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:           1;\n   2:           4,            5;\n   3:          28,           70,           43;\n   4:         280,         1050,         1290,           521;\n   5:        3640,        18200,        33540,         27092,          8113;\n   6:       58240,       364000,       894400,       1083680,        649040,        154001;\n   7:     1106560,      8299200,     25490400,      41179840,      36995280,      17556114,       3442915;\n   8:    24344320,    213012800,    785104320,    1585423840,    1899091040,    1351820778,     530208910,      88491173;\n   9:   608608000,   6086080000,  26170144000,   63416953600,   94954552000,   90121385200,   53020891000,   17698234600,   2568458401;\n  10: 17041024000, 191711520000, 942125184000, 2663512051200, 4785709420800, 5677647267600, 4453754844000, 2229977559600, 647251517052, 83033591453;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(12*n-28-(9*n^2-87*n+147)*x-(27*n^2-183*n+263)*x^2-(27*n^2-153*n+198)*x^3-(9*n^2-45*n+54)*x^4)*P(n-1,x) + (12-(18*n-108)*x-(72*n-306)*x^2-(108*n-390)*x^3-(72*n-234)*x^4-(18*n-54)*x^5)*d/dx P(n-1,x) + (9*x+45*x^2+90*x^3+90*x^4+45*x^5+9*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 28 - (9*n^2 - 87*n + 147)*x - (27*n^2 - 183*n + 263)*x^2 - (27*n^2 - 153*n + 198)*x^3 - (9*n^2 - 45*n + 54)*x^4)*P[n - 1, x] + (12 - (18*n - 108)*x - (72*n - 306)*x^2 - (108*n - 390)*x^3 - (72*n - 234)*x^4 - (18*n - 54)*x^5)*D[P[n - 1, x], x] + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(12*n - 28 - (9*n^2 - 87*n + 147)*x - (27*n^2 - 183*n + 263)*x^2 - (27*n^2 - 153*n + 198)*x^3 - (9*n^2 - 45*n + 54)*x^4)*v[i - 1] + (12 - (18*n - 108)*x - (72*n - 306)*x^2 - (108*n - 390)*x^3 - (72*n - 234)*x^4 - (18*n - 54)*x^5)*deriv(v[i - 1], x) + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e18d516553291356","steps":["x \u21a6 x/(x+1)"],"edges":["e-28b18fa0659c876e"],"text":"A290598 \u2192 x \u21a6 x/(x+1)"},"headline":"A290598 \u2192 x \u21a6 x/(x+1)","chain":null}