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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"b450067212b14af6f6e417c193b5c24843a8262ec1eca4abcbf26909d1d3b330","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-930be05ae3b321d9","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.","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;\n  2:       42,         18;\n  3:      393,        786,           81;\n  4:     3156,      17292,         9468,          324;\n  5:    23631,     284892,       427338,        94524,         1215;\n  6:   171006,    4031790,     12820140,      8063580,       855030,         4374;\n  7:  1216725,   52215966,    307208115,    409610820,    130539915,      7300350,       15309;\n  8:  8584872,  640078776,   6409572552,  15565211160,  10682620920,   1920236328,    60094104,     52488;\n  9: 60323739, 7573128600, 122066778708, 492299236968, 615374046210, 244133557416, 26505950100, 482589912, 177147;\n","recurrence_sentence":"The row polynomials satisfy (4*n-5)*P(n,x) = -(84*n-21+(96*n^2-240*n+54)*x+(64*n^3-240*n^2+236*n-45)*x^2)*P(n-2,x) - ((48*n-12)*x+(64*n^2-144*n+32)*x^2-(64*n^2-96*n+20)*x^3)*d/dx P(n-2,x) + (40*n-38+(32*n^2-80*n+50)*x)*P(n-1,x) + ((16*n-20)*x-(16*n-20)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(84*n - 21 + (96*n^2 - 240*n + 54)*x + (64*n^3 - 240*n^2 + 236*n - 45)*x^2)*P[n - 2, x] - ((48*n - 12)*x + (64*n^2 - 144*n + 32)*x^2 - (64*n^2 - 96*n + 20)*x^3)*D[P[n - 2, x], x] + (40*n - 38 + (32*n^2 - 80*n + 50)*x)*P[n - 1, x] + ((16*n - 20)*x - (16*n - 20)*x^2)*D[P[n - 1, x], x])/(4*n - 5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n - 21 + (96*n^2 - 240*n + 54)*x + (64*n^3 - 240*n^2 + 236*n - 45)*x^2)*v[i - 2] - ((48*n - 12)*x + (64*n^2 - 144*n + 32)*x^2 - (64*n^2 - 96*n + 20)*x^3)*deriv(v[i - 2], x) + (40*n - 38 + (32*n^2 - 80*n + 50)*x)*v[i - 1] + ((16*n - 20)*x - (16*n - 20)*x^2)*deriv(v[i - 1], x))/(4*n - 5)); 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-930be05ae3b321d9","steps":["D"],"edges":["e-275e089c88e33ae2"],"text":"A257621 \u2192 D"},"headline":"A257621 \u2192 D","chain":null}