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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-6938cf52b8fdd4be"]},{"id":168111,"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-6938cf52b8fdd4be"]},{"id":168112,"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-6938cf52b8fdd4be"]}],"region_exact":{},"incoming_edges":[{"id":"e-6938cf52b8fdd4be","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-f2d3e5cb5c029ac6","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":"f98f868971724fded101f8bc99ba49fa80da2b35e7d04d5ec6462c74f009bbf2","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,          4;\n  1:         8,         56,         112,           64;\n  2:        64,        640,        2368,         4096,          3328,           1024;\n  3:       512,       6656,       35328,        99840,        162816,         153600,          77824,           16384;\n  4:      4096,      65536,      450560,      1753088,       4263936,        6733824,        6914048,         4456448,         1638400,          262144;\n  5:     32768,     622592,     5242880,     25886720,      83394560,      184254464,      285147136,       309329920,       230686720,       112721920,         32505856,          4194304;\n  6:    262144,    5767168,    57409536,    342884352,    1372323840,     3889692672,     8037597184,     12266242048,     13831766016,     11387535360,       6656360448,       2617245696,        620756992,        67108864;\n  7:   2097152,   52428800,   601882624,   4213178368,   20126367744,    69539463168,   179610583040,    353267351552,    533645492224,    619322540032,     547859988480,     362891182080,     174281719808,     57310969856,     11542724608,      1073741824;\n  8:  16777216,  469762048,  6106906624,  48989470720,  271757344768,  1106759385088,  3427853664256,   8249424216064,  15621349703680,  23414349758464,   27784948744192,   25960929820672,   18869401944064,  10447507947520,   4256312590336,   1202590842880,   210453397504,    17179869184;\n  9: 134217728, 4160749568, 60397977600, 545997717504, 3447516561408, 16160082886656, 58333172072448, 165972031832064, 377802637443072, 694217978413056, 1034165579415552, 1249155804561408, 1218576711155712, 951820587368448, 586701122568192, 278932356071424, 98646808854528, 24429773979648, 3779571220480, 274877906944;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+24*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 24*x + 16*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 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 24*x + 16*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 A013611"],"edges":["e-6938cf52b8fdd4be"],"text":"A299989 \u2192 basis A013611"},"headline":"A299989 \u2192 basis A013611","chain":null}