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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"c8db17ce6cea151250d5f6d3365c3d0657e1fbd2b0339f39e061e9fd7acf155f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b362ab32ef269db1","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       7,        6;\n   3:      41,       78,        36;\n   4:     239,      708,       684,       216;\n   5:    1393,     5604,      8316,      5400,       1296;\n   6:    8119,    41274,     82836,     82080,      40176,       7776;\n   7:   47321,   290754,    736344,    984096,     732240,     287712,     46656;\n   8:  275807,  1987176,   6079752,  10240560,   10257840,    6111936,   2006208,    279936;\n   9: 1607521, 13287144,  47665224,  96937776,  122258160,   97930944,  48662208,  13716864,  1679616;\n  10: 9369319, 87380814, 359634456, 857377440, 1304917776, 1315022688, 877552704, 373994496, 92378880, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 7 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 7 + 6*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (6 + 6*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A013610","basis A013609"],"edges":["e-258aa4ea5bc7ae92","e-8548c09c5947adcc"],"text":"A111125 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A111125 \u2192 basis A013610 \u2192 basis A013609","chain":null}