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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":"83fee05d5dddf5f763d233f6d4ad3dec605ba5e49043be32f626c834cef9ef15","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-6a1d0871f44aed3c","role":"sequence","name":"basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by 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7:     576943600,     3050805840,     6738137696,      7996189724,      5439708753,      2076417208,      394024464,       25401600;\n  8:   42734614000,   266917880400,   715010755680,   1067480004892,    964154531793,    532973286648,   172339940368,    28494372096,    1625702400;\n  9: 3934757398000, 28379754610800, 89545912092360, 161656294743184, 183091256319351, 133906689132201, 62480123606440, 17550662666256, 2573033630976, 131681894400;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(2*n^6-8*n^5+14*n^4-12*n^3+4*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 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-e734a631f8279507","steps":["basis A007318","D+1"],"edges":["e-0b3719baa0dde23e","e-46c74c41479d6c2c"],"text":"A008955 \u2192 basis A007318 \u2192 D+1"},"headline":"A008955 \u2192 basis A007318 \u2192 D+1","chain":null}