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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":"ebbffeda7201acff6e3a44eaa09caa36e71d50d5110f23c3c2490910d22e5156","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-ee6acb764200c6c7","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) = 5*n + 2.","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:       32,         36,            4;\n  3:      252,        732,          268,            8;\n  4:     1860,      13556,        11388,         1908,           16;\n  5:    13292,     216500,       406480,       154660,        13420,           32;\n  6:    93748,    3116004,     11876400,      9499720,      1979580,        94068,          64;\n  7:   657964,   41984612,    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(250*n - 400)*x^4)*P[n - 2, x] - ((50*n - 80)*x + (125*n^2 - 575*n + 700)*x^2 - (125*n^2 - 900*n + 1470)*x^3 - (375*n - 1100)*x^4 - 250*x^5)*D[P[n - 2, x], x] + (45*n - 107 + (50*n^2 - 235*n + 363)*x + (125*n - 300)*x^2 + 50*x^3)*P[n - 1, x] + ((25*n - 65)*x - (25*n - 115)*x^2 - 50*x^3)*D[P[n - 1, x], x])/(5*n - 13 + 10*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n - 112 + (100*n^2 - 470*n + 636)*x + (125*n^3 - 850*n^2 + 2010*n - 1852)*x^2 + (375*n^2 - 1700*n + 1860)*x^3 + (250*n - 400)*x^4)*v[i - 2] - ((50*n - 80)*x + (125*n^2 - 575*n + 700)*x^2 - (125*n^2 - 900*n + 1470)*x^3 - (375*n - 1100)*x^4 - 250*x^5)*deriv(v[i - 2], x) + (45*n - 107 + (50*n^2 - 235*n + 363)*x + (125*n - 300)*x^2 + 50*x^3)*v[i - 1] + ((25*n - 65)*x - (25*n - 115)*x^2 - 50*x^3)*deriv(v[i - 1], x))/(5*n - 13 + 10*x)); 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-ee6acb764200c6c7","steps":["D+1"],"edges":["e-6065841777ea793c"],"text":"A257614 \u2192 D+1"},"headline":"A257614 \u2192 D+1","chain":null}