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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":"4ee3207e2476a0d042dbebcc3c5dfc1f3d464b9e9327c72c09f3904291f6ca5b","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-0c0e99d973615582","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*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:       1;\n  1:       6,         3;\n  2:      39,        48,          9;\n  3:     240,       639,        294,          27;\n  4:    1389,      7272,       6906,        1632,          81;\n  5:    7674,     72723,     130584,       62370,        8646,         243;\n  6:   41043,    660840,    2073147,     1828680,      511833,       44688,        729;\n  7:  214692,   5604351,   28800558,    43507725,    22181160,     3970953,     227814,      2187;\n  8: 1106265,  45217776,  362141748,   879096864,   758238510,   245715120,   29756964,   1152192,    6561;\n  9: 5642862, 351839619, 4226509248, 15668996244, 21476793852, 11686267698, 2562891216, 218093652, 5800326, 19683;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3+4*x)*P(n,x) = -(30*n-15+(24*n^2-60*n+84)*x+(8*n^3-28*n^2+82*n-107)*x^2+(24*n^2-56*n+46)*x^3+(16*n-8)*x^4)*P(n-2,x) - ((12*n-6)*x+(8*n^2-28*n+36)*x^2-(8*n^2-40*n+58)*x^3-(24*n-44)*x^4-16*x^5)*d/dx P(n-2,x) + (16*n-18+(8*n^2-20*n+44)*x+(20*n-26)*x^2+8*x^3)*P(n-1,x) + ((4*n-6)*x-(4*n-14)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 15 + (24*n^2 - 60*n + 84)*x + (8*n^3 - 28*n^2 + 82*n - 107)*x^2 + (24*n^2 - 56*n + 46)*x^3 + (16*n - 8)*x^4)*P[n - 2, x] - ((12*n - 6)*x + (8*n^2 - 28*n + 36)*x^2 - (8*n^2 - 40*n + 58)*x^3 - (24*n - 44)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (16*n - 18 + (8*n^2 - 20*n + 44)*x + (20*n - 26)*x^2 + 8*x^3)*P[n - 1, x] + ((4*n - 6)*x - (4*n - 14)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(2*n - 3 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n - 15 + (24*n^2 - 60*n + 84)*x + (8*n^3 - 28*n^2 + 82*n - 107)*x^2 + (24*n^2 - 56*n + 46)*x^3 + (16*n - 8)*x^4)*v[i - 2] - ((12*n - 6)*x + (8*n^2 - 28*n + 36)*x^2 - (8*n^2 - 40*n + 58)*x^3 - (24*n - 44)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (16*n - 18 + (8*n^2 - 20*n + 44)*x + (20*n - 26)*x^2 + 8*x^3)*v[i - 1] + ((4*n - 6)*x - (4*n - 14)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(2*n - 3 + 4*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-0c0e99d973615582","steps":["D+1"],"edges":["e-7d201c8b2a8a5c31"],"text":"A257611 \u2192 D+1"},"headline":"A257611 \u2192 D+1","chain":null}