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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"cef6f78151cdd7ad7d3f3dfd03b22bbd58144cd37e82f8811d1b343307b6c337","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5a70a6e97131d437","role":"sequence","name":"Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:     6,     21,      18;\n  1:    18,     99,     180,      108;\n  2:    36,    252,     657,      756,      324;\n  3:   108,    972,    3483,     6210,     5508,      1944;\n  4:   216,   2268,    9882,    22869,    29646,     20412,      5832;\n  5:   648,   8100,   43254,   127899,   226152,    239112,    139968,     34992;\n  6:  1296,  18144,  110808,   385560,   836001,   1156680,    997272,    489888,    104976;\n  7:  3888,  62208,  441288,  1821528,  4821363,   8486046,   9931896,   7453296,   3254256,    629856;\n  8:  7776, 136080, 1069200,  4966920, 15107310,  31436181,  45321930,  44702280,  28868400,  11022480,  1889568;\n  9: 23328, 454896, 4024080, 21315960, 75123450, 184952403, 324582876, 406038420, 354818880, 206277840, 71803584, 11337408;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+21*x+18*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 21*x + 18*x^2, P(1,x) = 18 + 99*x + 180*x^2 + 108*x^3.","mathematica":"P[0, x_] := 6 + 21*x + 18*x^2;\nP[1, x_] := 18 + 99*x + 180*x^2 + 108*x^3;\nP[n_, x_] := P[n, x] = Expand[(6 + 21*x + 18*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 6 + 21*x + 18*x^2; v[2] = 18 + 99*x + 180*x^2 + 108*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 21*x + 18*x^2)*v[i - 2]); 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-5a70a6e97131d437","steps":["basis A013610"],"edges":["e-f5362ff16cbef54d"],"text":"A064861 \u2192 basis A013610"},"headline":"A064861 \u2192 basis A013610","chain":null}