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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-f4ff934f738cae69"]},{"id":166182,"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-f4ff934f738cae69"]},{"id":166183,"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-f4ff934f738cae69"]}],"region_exact":{},"incoming_edges":[{"id":"e-f4ff934f738cae69","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-8d3e0db206d27b94","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"5b5760adc0836b8bf86675eab831ef2478a263b8b9c138f496f618c7e29869ae","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-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","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:             361,              315,                49;\n  1:            6859,            10717,              3822,               343;\n  2:          130321,           326655,            201145,             40474,              2401;\n  3:         2476099,          9405277,           8924370,           3026975,            396165,             16807;\n  4:        47045881,        261869895,         360341149,         183549590,          39808580,           3680733,           117649;\n  5:       893871739,       7137938437,       13724347182,        9864371643,        3155850390,         477419642,         32941720,           823543;\n  6:     16983563041,     191843501535,      502869026065,      490645295994,      217375569831,       47871613398,       5351382414,        286592964,         5764801;\n  7:    322687697779,    5106815981197,    17937582370890,    23145895022215,    13651168854015,     4106696112309,     661535621040,      56947998450,      2438510823,      40353607;\n  8:   6131066257801,  135036029395575,   627687930107749,  1051398877484830,   803626201294210,   317319772042791,   69100545669603,    8503213241880,    581524300875,   20378571535,    282475249;\n  9: 116490258898219, 3553854228089557, 21658953899324742, 46449770610147443, 45130223603221680, 22760651099370184, 6436371690145320, 1061922320135061, 103136815258785, 5741712971995, 167790297906, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (19+7*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 361 + 315*x + 49*x^2.","mathematica":"P[0, x_] := 361 + 315*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[(19 + 7*x)*P[n - 1, x] + 7*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 361 + 315*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 7*x)*v[i - 1] + 7*x*deriv(v[i - 1], x)); 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-5cf6acfdac607d6d","steps":["basis A008277"],"edges":["e-f4ff934f738cae69"],"text":"A038333 \u2192 basis A008277"},"headline":"A038333 \u2192 basis A008277","chain":null}