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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":"967847e0b47b506e343bcdaf3eda2b74502cb15016d78c93ac010aa40ff4638b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f91986a1e2f81fd1","role":"sequence","name":"basis_map_A013611 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","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:     1;\n  2:    11,      8;\n  3:    39,     96,      32;\n  4:   129,    560,     608,     128;\n  5:   393,   2464,    4896,    3328,      512;\n  6:  1147,   9432,   27616,   34432,    16896,     2048;\n  7:  3247,  33088,  128704,  238592,   214016,    81920,    8192;\n  8:  8993, 109408,  532032, 1319680,  1764352,  1228800,  385024,   32768;\n  9: 24497, 346560, 2024896, 6314496, 11335680, 11763712, 6676480, 1769472, 131072;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-5-(8*n^3-16*n^2-18*n+42)*x-(32*n^2-112*n+94)*x^2-(16*n^2-44*n+16)*x^3-(16*n-32)*x^4)*P(n,x) = -(8*n^3-24*n^2+8*n+5+(24*n^3-72*n^2-10*n+42)*x+(48*n^2-156*n+94)*x^2+(48*n^2-52*n+16)*x^3+(48*n-32)*x^4)*P(n-1,x) + (4*n^2-1+(32*n^2+16*n-10)*x+(72*n^2+136*n-24)*x^2+(32*n^2+352*n+44)*x^3+(272*n+228)*x^4+(64*n+224)*x^5+64*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 24*n^2 + 8*n + 5 + (24*n^3 - 72*n^2 - 10*n + 42)*x + (48*n^2 - 156*n + 94)*x^2 + (48*n^2 - 52*n + 16)*x^3 + (48*n - 32)*x^4)*P[n - 1, x] + (4*n^2 - 1 + (32*n^2 + 16*n - 10)*x + (72*n^2 + 136*n - 24)*x^2 + (32*n^2 + 352*n + 44)*x^3 + (272*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(-2*n + 5 + (8*n^3 - 16*n^2 - 18*n + 42)*x + (32*n^2 - 112*n + 94)*x^2 + (16*n^2 - 44*n + 16)*x^3 + (16*n - 32)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n^3 - 24*n^2 + 8*n + 5 + (24*n^3 - 72*n^2 - 10*n + 42)*x + (48*n^2 - 156*n + 94)*x^2 + (48*n^2 - 52*n + 16)*x^3 + (48*n - 32)*x^4)*v[i - 1] + (4*n^2 - 1 + (32*n^2 + 16*n - 10)*x + (72*n^2 + 136*n - 24)*x^2 + (32*n^2 + 352*n + 44)*x^3 + (272*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(-2*n + 5 + (8*n^3 - 16*n^2 - 18*n + 42)*x + (32*n^2 - 112*n + 94)*x^2 + (16*n^2 - 44*n + 16)*x^3 + (16*n - 32)*x^4)); 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-e073e5e0b021f36a","steps":["basis A013611","D+1"],"edges":["e-8337e61ed504a271","e-bf8fa3abda441fc6"],"text":"A035607 \u2192 basis A013611 \u2192 D+1"},"headline":"A035607 \u2192 basis A013611 \u2192 D+1","chain":null}