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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"5228ee6e6c39d4cf69b4061459bc83f0603f5b1a71dda6b9d7ff2bce82facb16","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-8cbd6010e0efad10","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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,         6;\n  2:       44,        96,         36;\n  3:      328,      1128,       1080,        216;\n  4:     2448,     11520,      18144,      10368,        1296;\n  5:    18272,    109152,     240192,     233280,       90720,        7776;\n  6:   136384,    987648,    2794176,    3856896,     2617920,      746496,      46656;\n  7:  1017984,   8662656,   29966976,   53758080,    52617600,    26780544,    5878656,     279936;\n  8:  7598336,  74293248,  303897600,  671625216,   862410240,   640991232,  256048128,   44789760,   1679616;\n  9: 56714752, 626443776, 2958280704, 7773677568, 12379309056, 12171617280, 7179798528, 2324588544, 332563968, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4+24*x+36*x^2)*P(n-2,x) + (8+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(4 + 24*x + 36*x^2)*P[n - 2, x] + (8 + 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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 24*x + 36*x^2)*v[i - 2] + (8 + 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-c30cb06ad75c5e21","steps":["basis A013609","basis A013610"],"edges":["e-49766d0fffe14f49","e-39656fe586dc4c6c"],"text":"A103327 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A103327 \u2192 basis A013609 \u2192 basis A013610","chain":null}