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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"6490af0e942f84042420cf496cde78a627d970bf2609993f8224ef494072bae3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-481b4b083ef06726","role":"sequence","name":"T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-481b4b083ef06726","role":"sequence","name":"T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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: 2,    30,      48,         8;\n  1: 2,    74,     308,       216,         16;\n  2: 2,   166,    1472,      2624,        928,          32;\n  3: 2,   354,    5972,     21744,      19968,        3872,           64;\n  4: 2,   734,   21936,    144280,     268960,      141216,        15872,          128;\n  5: 2,  1498,   75604,    830824,    2748944,     2960864,       950720,        64384,          256;\n  6: 2,  3030,  249792,   4347408,   23464704,    44754816,     30067200,      6187776,       259584,         512;\n  7: 2,  6098,  801908,  21269440,  176775488,   545107264,    652371584,    288169984,     39334400,     1042944,       1024;\n  8: 2, 12238, 2523632,  99129384, 1216905408,  5703569088,  11024025600,   8770811136,   2647122432,   245917184,    4182016,     2048;\n  9: 2, 24522, 7831988, 445769080, 7833600144, 53380281408, 156011794560, 201208317696, 110968120320, 23558001152, 1519420416, 16750592, 4096;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n+2)*x)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + (2*n+6)*x*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 30*x + 48*x^2 + 8*x^3, P(1,x) = 2 + 74*x + 308*x^2 + 216*x^3 + 16*x^4.","mathematica":"P[0, x_] := 2 + 30*x + 48*x^2 + 8*x^3;\nP[1, x_] := 2 + 74*x + 308*x^2 + 216*x^3 + 16*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n + 2)*x)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + (2*n + 6)*x*P[n - 1, x] + (x - 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 30*x + 48*x^2 + 8*x^3; v[2] = 2 + 74*x + 308*x^2 + 216*x^3 + 16*x^4; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n + 2)*x)*v[i - 2] + (x - 2*x^2)*deriv(v[i - 2], x) + (2*n + 6)*x*v[i - 1] + (x - 2*x^2)*deriv(v[i - 1], 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-481b4b083ef06726","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-00ad498f5c132fdf"],"text":"A156365 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A156365 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}