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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":"1ca2676020c6c99ac01952a424f52cdbc33e084999a6d2fd593740ca0ee433b2","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-3f0a1785078e99be","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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:         5;\n  2:        70,         50;\n  3:       715,       1430,         375;\n  4:      6380,      25740,       19140,         2500;\n  5:     52785,     372020,      558030,       211140,        15625;\n  6:    416370,    4720590,    12276660,      9441180,      2081850,        93750;\n  7:   3180215,   54976410,   227094765,    302793020,    137441025,     19081290,      546875;\n  8:  23745880,  602915000,  3735011640,   7872618520,   6225019400,   1808745000,   166221160,    3125000;\n  9: 174424285, 6328578440, 56459460540, 177044273560, 221305341950, 112918921080, 22150024540, 1395394280, 17578125;\n","recurrence_sentence":"The row polynomials satisfy (2*n+1)*P(n,x) = -(70*n+105+(40*n^2+40*n-30)*x+(8*n^3+12*n^2-2*n-3)*x^2)*P(n-2,x) - ((20*n+30)*x+(8*n^2-4*n-24)*x^2-(8*n^2+16*n+6)*x^3)*d/dx P(n-2,x) + (24*n+22+(8*n^2+8*n+2)*x)*P(n-1,x) + ((4*n+2)*x-(4*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 5;\nP[n_, x_] := P[n, x] = Expand[Together[(-(70*n + 105 + (40*n^2 + 40*n - 30)*x + (8*n^3 + 12*n^2 - 2*n - 3)*x^2)*P[n - 2, x] - ((20*n + 30)*x + (8*n^2 - 4*n - 24)*x^2 - (8*n^2 + 16*n + 6)*x^3)*D[P[n - 2, x], x] + (24*n + 22 + (8*n^2 + 8*n + 2)*x)*P[n - 1, x] + ((4*n + 2)*x - (4*n + 2)*x^2)*D[P[n - 1, x], x])/(2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n + 105 + (40*n^2 + 40*n - 30)*x + (8*n^3 + 12*n^2 - 2*n - 3)*x^2)*v[i - 2] - ((20*n + 30)*x + (8*n^2 - 4*n - 24)*x^2 - (8*n^2 + 16*n + 6)*x^3)*deriv(v[i - 2], x) + (24*n + 22 + (8*n^2 + 8*n + 2)*x)*v[i - 1] + ((4*n + 2)*x - (4*n + 2)*x^2)*deriv(v[i - 1], x))/(2*n + 1)); 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-3f0a1785078e99be","steps":["D"],"edges":["e-daa279775af9786b"],"text":"A257615 \u2192 D"},"headline":"A257615 \u2192 D","chain":null}