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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"3bec4abb02f1dab0ebd4f70beaeee6ca8ba0008606dbeaac9413c82175183aa4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b9dfb234567937e9","role":"sequence","name":"basis_map_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:      4,      1;\n  2:     17,      9,      1;\n  3:     68,     57,     14,      1;\n  4:    264,    307,    121,     19,      1;\n  5:   1002,   1500,    828,    209,     24,     1;\n  6:   3739,   6870,   4917,   1743,    321,    29,    1;\n  7:  13768,  30043,  26538,  12235,   3164,   457,   34,   1;\n  8:  50159, 126897, 133705,  76301,  25686,  5203,  617,  39,  1;\n  9: 181142, 521691, 639548, 436477, 182118, 48014, 7972, 801, 44, 1;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3-n^2-17*n-(3*n^3+2*n^2+11*n)*x-8*n^2*x^2-(n^2+7*n)*x^3-n*x^4)*P(n,x) = -(24*n^3+25*n^2-106*n-3+(12*n^3+56*n^2-98*n-49)*x+(35*n^2+19*n-102)*x^2+(4*n^2+26*n-70)*x^3+(3*n-15)*x^4-x^5)*P(n-1,x) + (39*n^2+143*n+104+(69*n^2+370*n+405)*x+(33*n^2+341*n+609)*x^2+(3*n^2+133*n+438)*x^3+(20*n+150)*x^4+(n+21)*x^5+x^6)*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[(-(24*n^3 + 25*n^2 - 106*n - 3 + (12*n^3 + 56*n^2 - 98*n - 49)*x + (35*n^2 + 19*n - 102)*x^2 + (4*n^2 + 26*n - 70)*x^3 + (3*n - 15)*x^4 - x^5)*P[n - 1, x] + (39*n^2 + 143*n + 104 + (69*n^2 + 370*n + 405)*x + (33*n^2 + 341*n + 609)*x^2 + (3*n^2 + 133*n + 438)*x^3 + (20*n + 150)*x^4 + (n + 21)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 + n^2 + 17*n + (3*n^3 + 2*n^2 + 11*n)*x + 8*n^2*x^2 + (n^2 + 7*n)*x^3 + n*x^4)]];\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] = (-(24*n^3 + 25*n^2 - 106*n - 3 + (12*n^3 + 56*n^2 - 98*n - 49)*x + (35*n^2 + 19*n - 102)*x^2 + (4*n^2 + 26*n - 70)*x^3 + (3*n - 15)*x^4 - x^5)*v[i - 1] + (39*n^2 + 143*n + 104 + (69*n^2 + 370*n + 405)*x + (33*n^2 + 341*n + 609)*x^2 + (3*n^2 + 133*n + 438)*x^3 + (20*n + 150)*x^4 + (n + 21)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 + n^2 + 17*n + (3*n^3 + 2*n^2 + 11*n)*x + 8*n^2*x^2 + (n^2 + 7*n)*x^3 + n*x^4)); 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-d5c14b9fec875ecc","steps":["x \u21a6 x+1","D+1"],"edges":["e-03e3acc3469b904b","e-52629bbb0f2f2329"],"text":"A121574 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A121574 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}