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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-42eb699b6c15b522"]},{"id":203908,"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-42eb699b6c15b522"]},{"id":203909,"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 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binomial(i,j)*11^(i-j)*12^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:                1;\n  1:               47,                72;\n  2:             2209,              6768,              5184;\n  3:           103823,            477144,            730944,             373248;\n  4:          4879681,          29901024,          68708736,           70170624,           26873856;\n  5:        229345007,        1756685160,        5382184320,         8245048320,         6315356160,          1934917632;\n  6:      10779215329,       99077043024,      379443994560,       775034542080,       890465218560,        545646772224,        139314069504;\n  7:     506623120463,     5432724525816,    24967414842048,     63746591086080,     97654352302080,      89758894030848,      45834328866816,     10030613004288;\n  8:   23811286661761,   291814917386688,  1564624663435008,   4793743649673216,   9179509116395520,   11249781385199616,    8616853826961408,   3771510489612288,    722204136308736;\n  9: 1119130473102767, 15429713756821128, 94548033233286912, 337958927301961728, 776586471247060992, 1189664381484859392, 1214976389601558528, 797674468552998912, 305492349658595328, 51998697814228992;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (47+72*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(47 + 72*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; for(n = 1, N, my(i = n + 1); v[i] = (47 + 72*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-9e472291b944d56e","steps":["basis A013609","basis A013610"],"edges":["e-15af34f262d68ea6","e-42eb699b6c15b522"],"text":"A038326 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038326 \u2192 basis A013609 \u2192 basis A013610","chain":null}