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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"36c31a049e7ec63dfd5122cd9bbe385304446d3cb41d1cb5ff507899fd8e56e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-cb5d49e71e46f2ed","role":"sequence","name":"Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","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-cb5d49e71e46f2ed","role":"sequence","name":"Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","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:       3,        5,         2;\n  2:       8,       22,        20,         6;\n  3:      30,      114,       162,       102,        24;\n  4:     144,      696,      1344,      1296,       624,       120;\n  5:     840,     4920,     12000,     15600,     11400,      4440,       720;\n  6:    5760,    39600,    116640,    190800,    187200,    110160,     36000,      5040;\n  7:   45360,   357840,   1234800,   2434320,   2998800,   2363760,   1164240,    327600,     40320;\n  8:  403200,  3588480,  14192640,  32739840,  48545280,  47980800,  31610880,  13386240,   3306240,   362880;\n  9: 3991680, 39553920, 176359680, 465937920, 807770880, 960180480, 792529920, 448519680, 166561920, 36650880, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n+1+n*x)*P(n,x) = (n^2+2*n+(2*n^2+3*n)*x+(n^2+n)*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[((n^2 + 2*n + (2*n^2 + 3*n)*x + (n^2 + n)*x^2)*P[n - 1, x])/(n + 1 + n*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] = ((n^2 + 2*n + (2*n^2 + 3*n)*x + (n^2 + n)*x^2)*v[i - 1])/(n + 1 + n*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-cb5d49e71e46f2ed","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-34b0075c2d0a2285"],"text":"A196347 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A196347 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}