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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"52b289252ed4807377d2eb33778bade4be95a733632d478c239da5000c3455a4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| 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-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| 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,   6,     4;\n  2: 2,  19,    53,      36;\n  3: 2,  44,   322,     856,       576;\n  4: 2,  85,  1296,    8465,     21652,      14400;\n  5: 2, 146,  4026,   52118,    317372,     787536,       518400;\n  6: 2, 231, 10465,  236093,   2771769,   16064776,     38920464,      25401600;\n  7: 2, 344, 23884,  860808,  17214626,  189013552,   1055469088,    2508535296,    1625702400;\n  8: 2, 489, 49368, 2668558,  83682534, 1541315217,  16105608920,   87348696336,  204385234176,   131681894400;\n  9: 2, 670, 94392, 7293340, 337823762, 9629524710, 166936414292, 1678585314880, 8892075327552, 20539316966400, 13168189440000;\n","recurrence_sentence":"The row polynomials satisfy (2+n^2*x)*P(n,x) = (2+(3*n^2+2*n+1)*x+(n^4+2*n^3+n^2)*x^2)*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[Together[((2 + (3*n^2 + 2*n + 1)*x + (n^4 + 2*n^3 + n^2)*x^2)*P[n - 1, x])/(2 + n^2*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] = ((2 + (3*n^2 + 2*n + 1)*x + (n^4 + 2*n^3 + n^2)*x^2)*v[i - 1])/(2 + 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-e734a631f8279507","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a0abdef027fe8e86"],"text":"A008955 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A008955 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}