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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":"669a340c4870c3b17e998505f1baca33ea6fc489ca0ef3b5caeb621cc5d7ae4c","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-7b068e1328e63a95","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+4x)^n.","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:       17,        12;\n  2:      145,       408,        144;\n  3:     1025,      5220,       7344,        1728;\n  4:     6625,     49200,     125280,      117504,        20736;\n  5:    40625,    397500,    1476000,     2505600,      1762560,       248832;\n  6:   240625,   2925000,   14310000,    35424000,     45100800,     25380864,      2985984;\n  7:  1390625,  20212500,  122850000,   400680000,    743904000,    757693440,    355332096,     35831808;\n  8:  7890625, 133500000,  970200000,  3931200000,   9616320000,  14282956800,  12123095040,   4873125888,   429981696;\n  9: 44140625, 852187500, 7209000000, 34927200000, 106142400000, 207712512000, 257093222400, 187042037760, 65787199488, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy (12*n-7+12*x)*P(n,x) = (60*n+25+(144*n+120)*x+144*x^2)*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[((60*n + 25 + (144*n + 120)*x + 144*x^2)*P[n - 1, x])/(12*n - 7 + 12*x)]];\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] = ((60*n + 25 + (144*n + 120)*x + 144*x^2)*v[i - 1])/(12*n - 7 + 12*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-5733d79f45a14084","steps":["basis A013611","D+1"],"edges":["e-1a3f71d8de8c31ec","e-46c1157f8d71ad24"],"text":"A013620 \u2192 basis A013611 \u2192 D+1"},"headline":"A013620 \u2192 basis A013611 \u2192 D+1","chain":null}