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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":"351ee1c8b6ebd212f5b57b3c5034acb573ea5e1b94fa1c76a48ff807d8fb9bcf","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:   1;\n  1:   3,    1;\n  2:   9,    7,    1;\n  3:  17,   26,   10,    1;\n  4:  35,   86,   59,   14,    1;\n  5:  49,  177,  203,   93,   17,    1;\n  6:  79,  399,  660,  465,  150,   21,    1;\n  7:  99,  616, 1389, 1455,  765,  201,   24,   1;\n  8: 141, 1108, 3262, 4584, 3375, 1344,  282,  28,  1;\n  9: 167, 1505, 5350, 9694, 9789, 5703, 1910, 350, 31, 1;\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) = 1, P(1,x) = 3 + x, P(2,x) = 9 + 7*x + x^2, P(3,x) = 17 + 26*x + 10*x^2 + x^3, P(4,x) = 35 + 86*x + 59*x^2 + 14*x^3 + x^4, P(5,x) = 49 + 177*x + 203*x^2 + 93*x^3 + 17*x^4 + x^5.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[2, x_] := 9 + 7*x + x^2;\nP[3, x_] := 17 + 26*x + 10*x^2 + x^3;\nP[4, x_] := 35 + 86*x + 59*x^2 + 14*x^3 + x^4;\nP[5, x_] := 49 + 177*x + 203*x^2 + 93*x^3 + 17*x^4 + x^5;\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] = 1; v[2] = 3 + x; v[3] = 9 + 7*x + x^2; v[4] = 17 + 26*x + 10*x^2 + x^3; v[5] = 35 + 86*x + 59*x^2 + 14*x^3 + x^4; v[6] = 49 + 177*x + 203*x^2 + 93*x^3 + 17*x^4 + x^5; 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+1"],"edges":["e-b1073907eb08b9b5","e-03d1f9e645566efe"],"text":"A026374 \u2192 D+1 \u2192 D+1"},"headline":"A026374 \u2192 D+1 \u2192 D+1","chain":null}