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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"d25237966797de3f2eac23bd9ed178e7b3917cd252732d195f0f08fbf418e7c7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"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: 2,       6;\n   2: 2,      54,         90;\n   3: 2,     282,       2304,          2160;\n   4: 2,    1230,      29646,        116748,          71280;\n   5: 2,    5058,     288180,       2906388,        6841908,          2993760;\n   6: 2,   20406,    2439378,      51048144,      287422992,        457131600,        152681760;\n   7: 2,   81834,   19145160,     739968264,     8127435024,      29611227672,      34389146160,        9160905600;\n   8: 2,  327582,  143787510,    9543015540,   182484708840,    1248961662504,    3221396103384,     2881145052000,      632102486400;\n   9: 2, 1310610, 1051522812,  114104709180,  3533641311780,   40645401319944,  191822672910240,   372022960709664,   266297285383200,    49303993939200;\n  10: 2, 5242758, 7564329018, 1296492735960, 61866777582360, 1113623591602824, 8622058665339576, 29994920973878256, 45668147294459520, 26933673382329600, 4289447472710400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(18*n-48)*x)*P(n-2,x) + (3*x-9*x^2)*d/dx P(n-2,x) + (18*n-12)*x*P(n-1,x) + (3*x-9*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 6*x, P(2,x) = 2 + 54*x + 90*x^2.","mathematica":"P[1, x_] := 2 + 6*x;\nP[2, x_] := 2 + 54*x + 90*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (18*n - 48)*x)*P[n - 2, x] + (3*x - 9*x^2)*D[P[n - 2, x], x] + (18*n - 12)*x*P[n - 1, x] + (3*x - 9*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 6*x; v[2] = 2 + 54*x + 90*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (18*n - 48)*x)*v[i - 2] + (3*x - 9*x^2)*deriv(v[i - 2], x) + (18*n - 12)*x*v[i - 1] + (3*x - 9*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7fdf5321df1ab451","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0f78bcf93daf4e45"],"text":"A290316 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A290316 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}