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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"9af9c1e61b774b248cb85676ae11ae93148d17749c4cb9a17db508565a44f4f6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-bac12d13055658fe","role":"sequence","name":"5-Stirling numbers of the second kind.","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-bac12d13055658fe","role":"sequence","name":"5-Stirling numbers of the second kind.","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:        6,        1;\n  1:       30,       12,        1;\n  2:      150,      102,       19,        1;\n  3:      750,      762,      235,       27,        1;\n  4:     3750,     5322,     2407,      451,       36,       1;\n  5:    18750,    35682,    22171,     6015,      775,      46,      1;\n  6:    93750,   232842,   190879,    70291,    12990,    1235,     57,     1;\n  7:   468750,  1490802,  1568995,   753207,   187201,   25340,   1862,    69,    1;\n  8:  2343750,  9413562, 12473767,  7594651,  2438016,  440601,  45822,  2690,   82,  1;\n  9: 11718750, 58825122, 96729931, 73230975, 29536795, 6844026, 944643, 78102, 3756, 96, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + x.","mathematica":"P[0, x_] := 6 + x;\nP[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + x*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] = 6 + x; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + x*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-bac12d13055658fe","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-76715f1adaa360d7"],"text":"A193685 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A193685 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}