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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"ea218a2ab868ad0b7495d1c89fe049dbd1217bf961f07e5fdf67ac4b3957ea8e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-8afa3bb9fc894abe","role":"sequence","name":"Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 3.","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-8afa3bb9fc894abe","role":"sequence","name":"Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 3.","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: -3, 1;\n 1: 16, -8, 1;\n 2: -100, 65, -14, 1;\n 3: 720, -564, 164, -21, 1;\n 4: -5880, 5306, -1891, 333, -29, 1;\n 5: 53760, -54272, 22504, -4910, 595, -38, 1;\n 6: -544320, 602424, -281382, 71959, -10890, 976, -48, 1;\n 7: 6048000, -7231200, 3722200, -1078196, 192374, -21665, 1505, -59, 1;\n 8: -73180800, 93485040, -52203556, 16737776, -3398199, 453369, -39774, 2214, -71, 1;\n 9: 958003200, -1296380160, 776159424, -270950064, 61118252, -9314676, 971985, -68616, 3138, -84, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2-x)*P(n,x) = -(n^2+6*n+9-(2*n+6)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -3 + x.","mathematica":"P[0, x_] := -3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 6*n + 9 - (2*n + 6)*x + x^2)*P[n - 1, x])/(n + 2 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -3 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 6*n + 9 - (2*n + 6)*x + x^2)*v[i - 1])/(n + 2 - 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-4f6e747b794bcc2e","steps":["x \u21a6 \u22122x+7","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-c581ca12cb2a386a","e-d08409290ded4df6"],"text":"A028338 \u2192 x \u21a6 \u22122x+7 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+7 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}