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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":"27f0ed330ab09e0f5af40d52d87fec8ee38f7b2ca76798b0d07e070056daac1f","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-c1d69b4cd7e1b71e","role":"sequence","name":"basis_map_A013609 of Pascal-(1,3,1) array.","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,       2;\n  2:     21,      22,       4;\n  3:     80,     156,      72,       8;\n  4:    305,     868,     720,     200,      16;\n  5:   1092,    4142,    5240,    2616,     512,      32;\n  6:   3829,   18186,   31236,   23864,    8304,    1248,     64;\n  7:  13120,   75032,  164016,  172592,   91648,   24192,   2944,    128;\n  8:  44289,  296488,  789184, 1067952,  784736,  314240,  66432,   6784,   256;\n  9: 147620, 1132986, 3561360, 5921040, 5632512, 3120960, 994176, 174720, 15360, 512;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+(4*n^3+4*n^2)*x+(14*n^2-8*n)*x^2+(2*n^2+11*n)*x^3+2*n*x^4)*P(n,x) = -(16*n^3-36*n^2-24*n+28+(28*n^3-28*n^2-132*n+92)*x+(82*n^2-144*n+22)*x^2+(14*n^2+43*n-121)*x^3+(10*n-44)*x^4-4*x^5)*P(n-1,x) + (16*n^2+16*n+(64*n^2+120*n+28)*x+(68*n^2+300*n+172)*x^2+(8*n^2+278*n+367)*x^3+(62*n+301)*x^4+(4*n+64)*x^5+4*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[(-(16*n^3 - 36*n^2 - 24*n + 28 + (28*n^3 - 28*n^2 - 132*n + 92)*x + (82*n^2 - 144*n + 22)*x^2 + (14*n^2 + 43*n - 121)*x^3 + (10*n - 44)*x^4 - 4*x^5)*P[n - 1, x] + (16*n^2 + 16*n + (64*n^2 + 120*n + 28)*x + (68*n^2 + 300*n + 172)*x^2 + (8*n^2 + 278*n + 367)*x^3 + (62*n + 301)*x^4 + (4*n + 64)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(4*n^2 + (4*n^3 + 4*n^2)*x + (14*n^2 - 8*n)*x^2 + (2*n^2 + 11*n)*x^3 + 2*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] = (-(16*n^3 - 36*n^2 - 24*n + 28 + (28*n^3 - 28*n^2 - 132*n + 92)*x + (82*n^2 - 144*n + 22)*x^2 + (14*n^2 + 43*n - 121)*x^3 + (10*n - 44)*x^4 - 4*x^5)*v[i - 1] + (16*n^2 + 16*n + (64*n^2 + 120*n + 28)*x + (68*n^2 + 300*n + 172)*x^2 + (8*n^2 + 278*n + 367)*x^3 + (62*n + 301)*x^4 + (4*n + 64)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(4*n^2 + (4*n^3 + 4*n^2)*x + (14*n^2 - 8*n)*x^2 + (2*n^2 + 11*n)*x^3 + 2*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-d8f9ad9ce0e039d0","steps":["basis A013609","D+1"],"edges":["e-471a582f60d8a81d","e-73908a48bb8ae4f1"],"text":"A081578 \u2192 basis A013609 \u2192 D+1"},"headline":"A081578 \u2192 basis A013609 \u2192 D+1","chain":null}