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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":"118e62781eed94f3e1a1b4fc28e41378773a525e20cac3ba6e6aa523ff2af45b","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-14e332ad5b6d2ede","role":"sequence","name":"Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.","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:  3,    2;\n  2:  7,   26,    21,      4;\n  3: 11,   82,   189,    164,     55,       6;\n  4: 15,  170,   693,   1284,   1155,     510,     105,        8;\n  5: 19,  290,  1725,   5156,   8415,    7734,    4025,     1160,      171,       10;\n  6: 23,  442,  3477,  14596,  35915,   53934,   50281,    29192,    10431,     2210,     253,      12;\n  7: 27,  626,  6141,  33444, 111815,  240486,  340473,   320648,   201267,    83610,   22517,    3756,     351,     14;\n  8: 31,  842,  9909,  66564, 283475,  805470, 1569001,  2125832,  2017287,  1342450,  623645,  199692,   42939,   5894,   465,   16;\n  9: 35, 1090, 14973, 119844, 622575, 2215830, 5568185, 10051720, 13163067, 12564650, 8750005, 4431660, 1618695, 419454, 74865, 8720, 595, 18;\n","recurrence_sentence":"The row polynomials satisfy -(16*n^3-44*n^2+38*n-10-(16*n^3-24*n^2+8*n)*x+(4*n^2-6*n+2)*x^2)*P(n,x) = -(16*n^3-28*n^2+14*n-2+(80*n^3-200*n^2+100*n-10)*x+(32*n^3-172*n^2+110*n+10)*x^2-(32*n^2-36*n-2)*x^3+4*n*x^4)*P(n-1,x) + ((8*n^2-6*n+1)*x+(56*n^2-44*n+5)*x^2+(56*n^2-56*n-5)*x^3+(8*n^2-20*n-1)*x^4-2*n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(16*n^3 - 28*n^2 + 14*n - 2 + (80*n^3 - 200*n^2 + 100*n - 10)*x + (32*n^3 - 172*n^2 + 110*n + 10)*x^2 - (32*n^2 - 36*n - 2)*x^3 + 4*n*x^4)*P[n - 1, x] + ((8*n^2 - 6*n + 1)*x + (56*n^2 - 44*n + 5)*x^2 + (56*n^2 - 56*n - 5)*x^3 + (8*n^2 - 20*n - 1)*x^4 - 2*n*x^5)*D[P[n - 1, x], x])/(-16*n^3 + 44*n^2 - 38*n + 10 + (16*n^3 - 24*n^2 + 8*n)*x - (4*n^2 - 6*n + 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n^3 - 28*n^2 + 14*n - 2 + (80*n^3 - 200*n^2 + 100*n - 10)*x + (32*n^3 - 172*n^2 + 110*n + 10)*x^2 - (32*n^2 - 36*n - 2)*x^3 + 4*n*x^4)*v[i - 1] + ((8*n^2 - 6*n + 1)*x + (56*n^2 - 44*n + 5)*x^2 + (56*n^2 - 56*n - 5)*x^3 + (8*n^2 - 20*n - 1)*x^4 - 2*n*x^5)*deriv(v[i - 1], x))/(-16*n^3 + 44*n^2 - 38*n + 10 + (16*n^3 - 24*n^2 + 8*n)*x - (4*n^2 - 6*n + 2)*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-cdb0db4ad473ee28","steps":["basis A008288","D"],"edges":["e-054f97e0823e3ab6","e-8695806e32ac2820"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 D"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 D","chain":null}