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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"d3d54f289c230d9242d2809df5899069ac69e6bb07e5636a0d03a3241281c5f8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-75a6475b3e851dc5","role":"sequence","name":"Triangle T(n,k) = n!*binomial(n-1, k-1) for 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-75a6475b3e851dc5","role":"sequence","name":"Triangle T(n,k) = n!*binomial(n-1, k-1) for 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:        3,         2;\n  1:        8,        14,          6;\n  2:       30,        84,         78,         24;\n  3:      144,       552,        792,        504,        120;\n  4:      840,      4080,       7920,       7680,       3720,         720;\n  5:     5760,     33840,      82800,     108000,      79200,       30960,       5040;\n  6:    45360,    312480,     922320,    1512000,    1486800,      876960,     287280,      40320;\n  7:   403200,   3185280,   11007360,   21732480,   26812800,    21168000,   10442880,    2943360,     362880;\n  8:  3991680,  35562240,  140797440,  325140480,  482630400,   477550080,  314979840,  133539840,   33022080,   3628800;\n  9: 43545600, 431827200, 1926892800, 5094835200, 8839756800, 10516262400, 8687347200, 4920652800, 1828915200, 402796800, 39916800;\n","recurrence_sentence":"The row polynomials satisfy (n+2+(n+1)*x)*P(n,x) = (n^2+4*n+3+(2*n^2+7*n+5)*x+(n^2+3*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.","mathematica":"P[0, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 + (2*n^2 + 7*n + 5)*x + (n^2 + 3*n + 2)*x^2)*P[n - 1, x])/(n + 2 + (n + 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 3 + (2*n^2 + 7*n + 5)*x + (n^2 + 3*n + 2)*x^2)*v[i - 1])/(n + 2 + (n + 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-75a6475b3e851dc5","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-107bbdcb60a21a66"],"text":"A156992 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A156992 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}