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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":"092eabcbdad3f225a88db134ef4b80b0d473483fa5205eaadf95192a94b2da9b","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-cffba19abf0eaf93","role":"sequence","name":"basis_map_A013611 of A Riordan array of coordination sequences.","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:     7,      4;\n  2:    31,     56,      16;\n  3:   109,    380,     336,       64;\n  4:   345,   1840,    3072,     1792,      256;\n  5:  1031,   7460,   19008,    20608,     8960,     1024;\n  6:  2967,  27208,   94416,   155392,   124160,    43008,     4096;\n  7:  8317,  92508,  408528,   915648,  1104640,   697344,   200704,    16384;\n  8: 22865, 299232, 1608832,  4596224,  7539200,  7151616,  3727360,   917504,   65536;\n  9: 61927, 932548, 5918208, 20625664, 43130368, 55511040, 43294720, 19202048, 4128768, 262144;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-18*n+5+(8*n^3-8*n^2-18*n-24)*x-(32*n+62)*x^2+(16*n^2-12*n-20)*x^3+(16*n+16)*x^4)*P(n,x) = -(8*n^3-56*n^2+72*n+11+(24*n^3-224*n^2+294*n+220)*x-(112*n^2-148*n-790)*x^2+(48*n^2-116*n+556)*x^3-(80*n+80)*x^4-128*x^5)*P(n-1,x) + (4*n^2-8*n-1+(32*n^2-64*n-26)*x+(72*n^2-136*n-160)*x^2+(32*n^2-308)*x^3+(144*n-44)*x^4+(64*n+160)*x^5+64*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[(-(8*n^3 - 56*n^2 + 72*n + 11 + (24*n^3 - 224*n^2 + 294*n + 220)*x - (112*n^2 - 148*n - 790)*x^2 + (48*n^2 - 116*n + 556)*x^3 - (80*n + 80)*x^4 - 128*x^5)*P[n - 1, x] + (4*n^2 - 8*n - 1 + (32*n^2 - 64*n - 26)*x + (72*n^2 - 136*n - 160)*x^2 + (32*n^2 - 308)*x^3 + (144*n - 44)*x^4 + (64*n + 160)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(8*n^2 - 18*n + 5 + (8*n^3 - 8*n^2 - 18*n - 24)*x - (32*n + 62)*x^2 + (16*n^2 - 12*n - 20)*x^3 + (16*n + 16)*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] = (-(8*n^3 - 56*n^2 + 72*n + 11 + (24*n^3 - 224*n^2 + 294*n + 220)*x - (112*n^2 - 148*n - 790)*x^2 + (48*n^2 - 116*n + 556)*x^3 - (80*n + 80)*x^4 - 128*x^5)*v[i - 1] + (4*n^2 - 8*n - 1 + (32*n^2 - 64*n - 26)*x + (72*n^2 - 136*n - 160)*x^2 + (32*n^2 - 308)*x^3 + (144*n - 44)*x^4 + (64*n + 160)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(8*n^2 - 18*n + 5 + (8*n^3 - 8*n^2 - 18*n - 24)*x - (32*n + 62)*x^2 + (16*n^2 - 12*n - 20)*x^3 + (16*n + 16)*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-ec35d1145e9a1211","steps":["basis A013611","D+1"],"edges":["e-14cc72d2ba3c3aa2","e-e2a70a1644728402"],"text":"A113413 \u2192 basis A013611 \u2192 D+1"},"headline":"A113413 \u2192 basis A013611 \u2192 D+1","chain":null}