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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":"aea7c9984c14f63e913fd02d86f2619e675d9ab10fda2d562937872b8c25a614","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-44fd873021acce78","role":"sequence","name":"basis_map_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:      7,       4;\n  2:     38,      60,       16;\n  3:    173,     492,      368,       64;\n  4:    729,    3052,     4096,     1984,       256;\n  5:   2928,   16296,    32400,    27968,      9984,      1024;\n  6:  11389,   79320,   211008,   269760,    170496,     48128,     4096;\n  7:  43291,  362596,  1213584,  2080192,   1942784,    965632,   225280,    16384;\n  8: 161726, 1584084,  6404128, 13846592,  17339136,  12700672,  5193728,  1032192,   65536;\n  9: 596057, 6687396, 31729808, 83075392, 131347968, 128884736, 77467648, 26886144, 4653056, 262144;\n","recurrence_sentence":"The row polynomials satisfy -(192*n^3-688*n^2+100*n-(768*n^3-64*n^2+800*n)*x-(1280*n^2-1536*n)*x^2-(1024*n^2-512*n)*x^3-1024*n*x^4)*P(n,x) = -(1536*n^3-3392*n^2-1648*n+732+(3072*n^3-11008*n^2-10304*n+6128)*x-(1024*n^2+16640*n-13440)*x^2+(4096*n^2-7168*n+512)*x^3-9216*x^4-4096*x^5)*P(n-1,x) + (624*n^2+260*n-91+(4416*n^2+3088*n-696)*x+(8448*n^2+13184*n-144)*x^2+(3072*n^2+24064*n+9984)*x^3+(17408*n+25344)*x^4+(4096*n+18432)*x^5+4096*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[(-(1536*n^3 - 3392*n^2 - 1648*n + 732 + (3072*n^3 - 11008*n^2 - 10304*n + 6128)*x - (1024*n^2 + 16640*n - 13440)*x^2 + (4096*n^2 - 7168*n + 512)*x^3 - 9216*x^4 - 4096*x^5)*P[n - 1, x] + (624*n^2 + 260*n - 91 + (4416*n^2 + 3088*n - 696)*x + (8448*n^2 + 13184*n - 144)*x^2 + (3072*n^2 + 24064*n + 9984)*x^3 + (17408*n + 25344)*x^4 + (4096*n + 18432)*x^5 + 4096*x^6)*D[P[n - 1, x], x])/(-192*n^3 + 688*n^2 - 100*n + (768*n^3 - 64*n^2 + 800*n)*x + (1280*n^2 - 1536*n)*x^2 + (1024*n^2 - 512*n)*x^3 + 1024*n*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] = (-(1536*n^3 - 3392*n^2 - 1648*n + 732 + (3072*n^3 - 11008*n^2 - 10304*n + 6128)*x - (1024*n^2 + 16640*n - 13440)*x^2 + (4096*n^2 - 7168*n + 512)*x^3 - 9216*x^4 - 4096*x^5)*v[i - 1] + (624*n^2 + 260*n - 91 + (4416*n^2 + 3088*n - 696)*x + (8448*n^2 + 13184*n - 144)*x^2 + (3072*n^2 + 24064*n + 9984)*x^3 + (17408*n + 25344)*x^4 + (4096*n + 18432)*x^5 + 4096*x^6)*deriv(v[i - 1], x))/(-192*n^3 + 688*n^2 - 100*n + (768*n^3 - 64*n^2 + 800*n)*x + (1280*n^2 - 1536*n)*x^2 + (1024*n^2 - 512*n)*x^3 + 1024*n*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-d5c14b9fec875ecc","steps":["basis A013611","D+1"],"edges":["e-3c5bbf704ea0dfe8","e-ee594bf224798798"],"text":"A121574 \u2192 basis A013611 \u2192 D+1"},"headline":"A121574 \u2192 basis A013611 \u2192 D+1","chain":null}