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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":"eac693e1fe4a1ae15e7086c2f4c2c225462a6c63dd43843e161386685c0d2d44","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-582f0cc7fb344588","role":"sequence","name":"one_plus_derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.","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:   0;\n  1:   1;\n  2:   5,    2;\n  3:  12,   14,    3;\n  4:  28,   58,   30,    4;\n  5:  41,  136,  135,   48,    5;\n  6:  69,  330,  495,  300,   75,    6;\n  7:  88,  528, 1125, 1080,  495,  102,    7;\n  8: 128,  980, 2772, 3660, 2460,  852,  140,   8;\n  9: 153, 1352, 4674, 8136, 7755, 4152, 1218, 176, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+9*x+30*x^2+45*x^3+30*x^4+9*x^5+x^6)*P(n-6,x) - (3+18*x+33*x^2+18*x^3+3*x^4)*P(n-4,x) + (3+9*x+3*x^2)*P(n-2,x) for n >= 6, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 12 + 14*x + 3*x^2, P(4,x) = 28 + 58*x + 30*x^2 + 4*x^3, P(5,x) = 41 + 136*x + 135*x^2 + 48*x^3 + 5*x^4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 5 + 2*x;\nP[3, x_] := 12 + 14*x + 3*x^2;\nP[4, x_] := 28 + 58*x + 30*x^2 + 4*x^3;\nP[5, x_] := 41 + 136*x + 135*x^2 + 48*x^3 + 5*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*P[n - 6, x] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*P[n - 4, x] + (3 + 9*x + 3*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 5 + 2*x; v[4] = 12 + 14*x + 3*x^2; v[5] = 28 + 58*x + 30*x^2 + 4*x^3; v[6] = 41 + 136*x + 135*x^2 + 48*x^3 + 5*x^4; for(n = 6, N, my(i = n + 1); v[i] = (1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*v[i - 6] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*v[i - 4] + (3 + 9*x + 3*x^2)*v[i - 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-4e0b19699f4cf0db","steps":["D+1","D"],"edges":["e-b1073907eb08b9b5","e-2b97365be7b23e77"],"text":"A026374 \u2192 D+1 \u2192 D"},"headline":"A026374 \u2192 D+1 \u2192 D","chain":null}