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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"b64cff3af82a8122761b1fd71ed975137e08f4442010e4174bd7247e090b6cc6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-677d34c7af372519","role":"sequence","name":"Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.","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-677d34c7af372519","role":"sequence","name":"Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.","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,    1;\n  1: 2,    4,     2;\n  2: 2,   10,    14,       6;\n  3: 2,   22,    62,      66,      24;\n  4: 2,   46,   230,     450,     384,      120;\n  5: 2,   94,   782,    2490,    3720,     2640,      720;\n  6: 2,  190,  2534,   12306,   28560,    34440,    20880,     5040;\n  7: 2,  382,  7982,   56826,  192024,   349440,   352800,   186480,    40320;\n  8: 2,  766, 24710,  251250, 1187424,  3056760,  4566240,  3961440,  1854720,   362880;\n  9: 2, 1534, 75662, 1079130, 6942120, 24277680, 50304240, 63655200, 48384000, 20321280, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (x + 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 + x; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (x + 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-677d34c7af372519","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-766d0e8c11bad8cd"],"text":"A028246 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028246 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}