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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e154cc1df6252a67eeff40c2bf470df085529670c37cc9dfb7c4cefb8fee12d0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0552ada0abe44307","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:       1;\n  1:       6,        12;\n  2:      35,       144,        144;\n  3:     205,      1272,       2592,       1728;\n  4:    1200,      9960,      30672,      41472,       20736;\n  5:    7025,     73020,     301104,     615168,      622080,      248832;\n  6:   41125,    513600,    2654640,    7264512,    11093760,     8957952,     2985984;\n  7:  240750,   3510900,   21818160,   74867328,   153135360,   186624000,   125411328,    35831808;\n  8: 1409375,  23505000,  170658000,  704332800,  1806727680,  2948659200,  2988969984,  1719926784,   429981696;\n  9: 8250625, 154881000, 1286571600, 6205714560, 19149716736, 39194772480, 53195304960, 46151368704, 23219011584, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+12*x)*P(n-2,x) + (5+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 12*x;\nP[n_, x_] := P[n, x] = Expand[(5 + 12*x)*P[n - 2, x] + (5 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (5 + 12*x)*v[i - 2] + (5 + 12*x)*v[i - 1]); 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-21854f4257effb01","steps":["basis A013611","basis A013610"],"edges":["e-3723cb941a96ee35","e-8244d295c87c92ae"],"text":"A030528 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A030528 \u2192 basis A013611 \u2192 basis A013610","chain":null}
