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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-1cab7244184e4365"]},{"id":166734,"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-1cab7244184e4365"]},{"id":166735,"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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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,          2;\n  1:         8,         28,          28,           8;\n  2:        64,        320,         592,         512,          208,           32;\n  3:       512,       3328,        8832,       12480,        10176,         4800,         1216,           128;\n  4:      4096,      32768,      112640,      219136,       266496,       210432,       108032,         34816,          6400,           512;\n  5:     32768,     311296,     1310720,     3235840,      5212160,      5757952,      4455424,       2416640,        901120,        220160,         31744,         2048;\n  6:    262144,    2883584,    14352384,    42860544,     85770240,    121552896,    125587456,      95830016,      54030336,      22241280,       6500352,      1277952,       151552,         8192;\n  7:   2097152,   26214400,   150470656,   526647296,   1257897984,   2173108224,   2806415360,    2759901184,    2084552704,    1209614336,     535019520,    177192960,     42549248,      6995968,      704512,      32768;\n  8:  16777216,  234881024,  1526726656,  6123683840,  16984834048,  34586230784,  53560213504,   64448626688,   61020897280,   45731151872,   27133739008,  12676235264,   4606787584,   1275330560,   259784704,   36700160,    3211264,    131072;\n  9: 134217728, 2080374784, 15099494400, 68249714688, 215469785088, 505002590208, 911455813632, 1296656498688, 1475791552512, 1355894489088, 1009927323648, 609939357696, 297504079872, 116189036544, 35809394688, 8512339968, 1505230848, 186384384, 14417920, 524288;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+12*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 12*x + 4*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 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 12*x + 4*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 A013609"],"edges":["e-1cab7244184e4365"],"text":"A299989 \u2192 basis A013609"},"headline":"A299989 \u2192 basis A013609","chain":null}