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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-03b8863ee0776575"]},{"id":168292,"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-03b8863ee0776575"]},{"id":168293,"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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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:             529,              2024,               1936;\n  1:           12167,             69828,             133584,               85184;\n  2:          279841,           2141392,            6144864,             7836928,             3748096;\n  3:         6436343,          61565020,          235553120,           450623360,           431031040,            164916224;\n  4:       148035889,        1699194552,         8126582640,         20728674560,         29741141760,          22758438912,           7256313856;\n  5:      3404825447,       45595053812,       261675961008,        834329151040,       1596107941120,        1832054332416,        1168266530816,         319277809664;\n  6:     78310985281,     1198498557344,      8024729470912,      30703312758272,      73420965291520,      112365999054848,      107480520835072,       58747116978176,       14048223625216;\n  7:   1801152661463,    31011150171276,    237302714354112,    1059264290160384,    3039627963068928,     5814940451088384,     7416155937619968,     6080326607241216,     2907982290419712,      618121839509504;\n  8:  41426511213649,   792507171043720,   6822453037680720,   34804398105269760,  116519071917642240,   267487260750065664,   426428966413148160,   466158373221826560,   334417963398266880,   142168023087185920,   27197360938418176;\n  9: 952809757913927, 20050431427406116, 191786735392580240, 1100689090079156160, 4211332170737640960, 11279046161627768832, 21577305700505296896, 29484517106280529920, 28202581579920506880, 17984254920529018880, 6880932317419798528, 1196683881290399744;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 2024*x + 1936*x^2.","mathematica":"P[0, x_] := 529 + 2024*x + 1936*x^2;\nP[n_, x_] := P[n, x] = Expand[(23 + 44*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] = 529 + 2024*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 44*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-344e4bfa9933088a","steps":["basis A013611"],"edges":["e-03b8863ee0776575"],"text":"A038337 \u2192 basis A013611"},"headline":"A038337 \u2192 basis A013611","chain":null}