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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":"de7cdc6442d16113c2ff814d6e983b6c45ad59ca9bbb60b5563a4114d9170137","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4d7e6ff489a3ffc4","role":"sequence","name":"basis_map_A013609 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 peaks (1 <= k <= n).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       5,        2;\n   3:      27,       24,        4;\n   4:     147,      222,       84,        8;\n   5:     805,     1776,     1160,      256,       16;\n   6:    4433,    13050,    12680,     4880,      720,       32;\n   7:   24535,    90936,   119820,    69440,    18000,     1920,      64;\n   8:  136391,   611702,  1028412,   817880,   322000,    60704,    4928,    128;\n   9:  761097,  4014240,  8255632,  8465408,  4614624,  1329664,  191744,  12288,   256;\n  10: 4261149, 25870770, 63127440, 79890720, 56619360, 22744512, 5037312, 576000, 29952, 512;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-4*n^2-n+7+(8*n^2-8*n-16)*x+(4*n+4)*x^2)*P(n,x) = (20*n^3-20*n^2+13*n-1+(24*n^3-16*n^2-46*n+6)*x+(48*n^2-100*n+68)*x^2+(24*n-24)*x^3)*P(n-1,x) + (4*n^2-2*n+1+(8*n-8)*x-(16*n^2-8*n)*x^2-(32*n-32)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((20*n^3 - 20*n^2 + 13*n - 1 + (24*n^3 - 16*n^2 - 46*n + 6)*x + (48*n^2 - 100*n + 68)*x^2 + (24*n - 24)*x^3)*P[n - 1, x] + (4*n^2 - 2*n + 1 + (8*n - 8)*x - (16*n^2 - 8*n)*x^2 - (32*n - 32)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(4*n^3 - 4*n^2 - n + 7 + (8*n^2 - 8*n - 16)*x + (4*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n^3 - 20*n^2 + 13*n - 1 + (24*n^3 - 16*n^2 - 46*n + 6)*x + (48*n^2 - 100*n + 68)*x^2 + (24*n - 24)*x^3)*v[i - 1] + (4*n^2 - 2*n + 1 + (8*n - 8)*x - (16*n^2 - 8*n)*x^2 - (32*n - 32)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(4*n^3 - 4*n^2 - n + 7 + (8*n^2 - 8*n - 16)*x + (4*n + 4)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["basis A013609","D+1"],"edges":["e-077b645e97fdde1b","e-856e1e31ef09bb6f"],"text":"A114656 \u2192 basis A013609 \u2192 D+1"},"headline":"A114656 \u2192 basis A013609 \u2192 D+1","chain":null}