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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"fa567a65d40ee8fe8f7a15986c198a7f4d800816172b34f037eecfb74448f72b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fac4d4a000d744e8","role":"sequence","name":"Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:    2,     3;\n  2:    5,    15,     11;\n  3:   13,    59,     88,      43;\n  4:   34,   207,    468,     465,     171;\n  5:   89,   680,   2063,    3104,    2314,     683;\n  6:  233,  2142,   8157,   16462,   18559,   11075,    2731;\n  7:  610,  6555,  30042,   76095,  115005,  103665,   51580,   10923;\n  8: 1597, 19641, 105244,  320837,  608455,  734861,  551798,  235445,   43691;\n  9: 4181, 57913, 355235, 1266248, 2889997, 4378793, 4403439, 2833422, 1058286, 174763;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (13*n-11+(59*n-52)*x+(97*n-89)*x^2+(69*n-66)*x^3+(18*n-18)*x^4)*P(n-1,x) - (5+34*x+90*x^2+116*x^3+73*x^4+18*x^5)*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[((13*n - 11 + (59*n - 52)*x + (97*n - 89)*x^2 + (69*n - 66)*x^3 + (18*n - 18)*x^4)*P[n - 1, x] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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] = ((13*n - 11 + (59*n - 52)*x + (97*n - 89)*x^2 + (69*n - 66)*x^3 + (18*n - 18)*x^4)*v[i - 1] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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-fac4d4a000d744e8","steps":["x \u21a6 x/(x+1)"],"edges":["e-2d8eb3c5691a59ac"],"text":"A060920 \u2192 x \u21a6 x/(x+1)"},"headline":"A060920 \u2192 x \u21a6 x/(x+1)","chain":null}