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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":"ece643d07c1b3ebcbe92f5a74e36e01bded7ff7c94cf3d4a700e3238d6646cb5","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-8b333ce67e8457dc","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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:       11,         8;\n  2:       67,       184,         64;\n  3:      381,      1872,       2304,        512;\n  4:     2133,     15248,      34304,      25600,        4096;\n  5:    11903,    112120,     373120,     517120,      266240,       32768;\n  6:    66463,    778344,    3453120,    7175680,     6942720,     2654208,      262144;\n  7:   371801,   5210912,   28987392,   81105920,   118988800,    86245376,    25690112,     2097152;\n  8:  2084521,  34036768,  228057088,  806055936,  1605976064,  1781792768,  1012924416,   243269632,   16777216;\n  9: 11713203, 218420712, 1713574656, 7336734720, 18586681344, 28216000512, 24750587904, 11400118272, 2264924160, 134217728;\n","recurrence_sentence":"The row polynomials satisfy (64*n^3+96*n^2-78*n-28+(128*n^2+152*n-208)*x+(64*n+128)*x^2)*P(n,x) = (448*n^3+288*n^2-114*n-28+(1536*n^3+1152*n^2-1256*n-208)*x+(3072*n^2-512*n+128)*x^2+1536*n*x^3)*P(n-1,x) - (32*n^2+32*n+5+(384*n^2+448*n+72)*x+(1024*n^2+1792*n+336)*x^2+(2048*n+768)*x^3+1024*x^4)*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[((448*n^3 + 288*n^2 - 114*n - 28 + (1536*n^3 + 1152*n^2 - 1256*n - 208)*x + (3072*n^2 - 512*n + 128)*x^2 + 1536*n*x^3)*P[n - 1, x] - (32*n^2 + 32*n + 5 + (384*n^2 + 448*n + 72)*x + (1024*n^2 + 1792*n + 336)*x^2 + (2048*n + 768)*x^3 + 1024*x^4)*D[P[n - 1, x], x])/(64*n^3 + 96*n^2 - 78*n - 28 + (128*n^2 + 152*n - 208)*x + (64*n + 128)*x^2)]];\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] = ((448*n^3 + 288*n^2 - 114*n - 28 + (1536*n^3 + 1152*n^2 - 1256*n - 208)*x + (3072*n^2 - 512*n + 128)*x^2 + 1536*n*x^3)*v[i - 1] - (32*n^2 + 32*n + 5 + (384*n^2 + 448*n + 72)*x + (1024*n^2 + 1792*n + 336)*x^2 + (2048*n + 768)*x^3 + 1024*x^4)*deriv(v[i - 1], x))/(64*n^3 + 96*n^2 - 78*n - 28 + (128*n^2 + 152*n - 208)*x + (64*n + 128)*x^2)); 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-8ce61d302cec295b","steps":["basis A013611","D+1"],"edges":["e-034e0d2a0f7f14a7","e-57aa2dad595e092f"],"text":"A114687 \u2192 basis A013611 \u2192 D+1"},"headline":"A114687 \u2192 basis A013611 \u2192 D+1","chain":null}