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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-78fc61375bb1ec88"]},{"id":167426,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-78fc61375bb1ec88"]},{"id":167427,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-78fc61375bb1ec88"]}],"region_exact":{},"incoming_edges":[{"id":"e-78fc61375bb1ec88","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-8624395344cd3600","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"46f61eaba3e8035c821398d93801ecc95fcb0cf0ec5ebed5ceb80194ffc044b6","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-7e353d0275e3ebc5","role":"sequence","name":"Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.","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,          3;\n  1:         8,         42,          63,           27;\n  2:        64,        480,        1332,         1728,          1053,           243;\n  3:       512,       4992,       19872,        42120,         51516,         36450,          13851,           2187;\n  4:      4096,      49152,      253440,       739584,       1349136,       1597968,        1230552,         594864,         164025,          19683;\n  5:     32768,     466944,     2949120,     10920960,      26386560,      43724448,       50750064,       41290560,       23094720,        8463690,        1830519,         177147;\n  6:    262144,    4325376,    32292864,    144654336,     434211840,     923042304,     1430519616,     1637345664,     1384738416,      855029520,      374843052,      110539728,       19663317,        1594323;\n  7:   2097152,   39321600,   338558976,   1777434624,    6368108544,   16502040576,    31966824960,    47155499136,    53424805824,    46501638624,    30851921520,    15326758440,     5520609108,     1361551842,      205667667,      14348907;\n  8:  16777216,  352321536,  3435134976,  20667432960,   85985722368,  262639190016,   610084306944,  1101165207552,  1563898855680,  1758059106048,  1564668119808,  1096463402496,   597713818464,   248204204640,    75838756464,   16070775840,   2109289329,    129140163;\n  9: 134217728, 3120562176, 33973862400, 230342787072, 1090815787008, 3834863419392, 10382051377152, 22154591895552, 37822923343872, 52125139118592, 58237498568448, 52758265330944, 38600064870912, 22612652991936, 10453810101408, 3727501664832, 988697087928, 183637311786, 21308126895, 1162261467;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+18*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 18*x + 9*x^2)*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 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 18*x + 9*x^2)*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-7e353d0275e3ebc5","steps":["basis A013610"],"edges":["e-78fc61375bb1ec88"],"text":"A299989 \u2192 basis A013610"},"headline":"A299989 \u2192 basis A013610","chain":null}