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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-5da9b7e6b9d7c34c"]},{"id":198224,"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-5da9b7e6b9d7c34c"]},{"id":198225,"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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2)","lean_route":null,"lean_declarations":[],"certificate_hash":"76a590c4053abbdfedd8b23e7f7d64e8abb5fcc886a5581fc4f95c16332d2ea2","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-f96c16e0c5fd8334","role":"sequence","name":"basis_map_A013610 of 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:               1600,                 2121,                   441;\n  1:              64000,               162981,                 80703,                   9261;\n  2:            2560000,             11285841,              10040247,                2648646,                 194481;\n  3:          102400000,            742196301,            1060302915,              483655725,               79737210,                4084101;\n  4:         4096000000,          47424374361,          102530961351,            72082900890,            20044184265,             2266676055,              85766121;\n  5:       163840000000,        2978902836021,         9403450692363,          9577688980041,          3999219767550,           749595897540,           61837373241,            1801088541;\n  6:      6553600000000,      185153712997281,       833639916330207,       1183974429483846,        697034719757061,        192675020261850,        26006517806346,         1635388395228,          37822859361;\n  7:    262144000000000,    11432002092834141,     72246701112019875,     139455804479770485,     111295768269036330,      42575607052866531,      8363257381352286,       851954503840902,       42210311046876,       794280046581;\n  8:  10485760000000000,   702857151662882601,   6164301535135146711,   15881128584768777330,   16729247159435685105,    8510674156315409925,   2282388473414676627,    334943897226646680,    26670789278409150,   1068306662651445,    16679880978201;\n  9: 419430400000000000, 43094487211435838661, 520232726066002564923, 1765206576469022145921, 2407930348050169276950, 1585361943013883826330, 557600643869459928507, 110564666723091138327, 12581346011668683480, 804728800593773055, 26604410160230595, 350277500542221;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (40+21*x)*P(n-1,x) + 21*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1600 + 2121*x + 441*x^2.","mathematica":"P[0, x_] := 1600 + 2121*x + 441*x^2;\nP[n_, x_] := P[n, x] = Expand[(40 + 21*x)*P[n - 1, x] + 21*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] = 1600 + 2121*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = (40 + 21*x)*v[i - 1] + 21*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 A013610","basis A008277"],"edges":["e-065a527a7a0e90c1","e-5da9b7e6b9d7c34c"],"text":"A038333 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A038333 \u2192 basis A013610 \u2192 basis A008277","chain":null}