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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-50822203bd5493c4"]},{"id":167604,"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-50822203bd5493c4"]},{"id":167605,"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,              1518,               1089;\n  1:           12167,             52371,              75141,              35937;\n  2:          279841,           1606044,            3456486,            3306204,             1185921;\n  3:         6436343,          46173765,          132498630,          190106730,           136380915,            39135393;\n  4:       148035889,        1274395914,         4571202735,         8744909580,          9410283135,          5400684234,          1291467969;\n  5:      3404825447,       34196290359,       147192728067,       351982610595,        505018528245,        434755080837,        207926343009,         42618442977;\n  6:     78310985281,      898873918008,      4513910327388,     12952960069896,      23230852299270,      26664978291336,      19129223556828,       7841793507768,       1406408618241;\n  7:   1801152661463,    23258362628457,    133482776824188,    446877122411412,     961757285189778,    1379912626576638,    1319916425421132,     811625628053988,     291126583975887,      46411484401953;\n  8:  41426511213649,   594380378282790,   3837629833695405,  14683105450660680,   36867362598941490,   63475980822525348,   75895194461715090,   62224631484139080,   33479557157227005,   10674641412449190,   1531578985264449;\n  9: 952809757913927, 15037823570554587, 107880038658326385, 464353209877144005, 1332491819647456710, 2676570524683152174, 3840296839762783554, 3935707941371796810, 2823442653592810755, 1350342138674822535, 387489483271905597, 50542106513726817;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+33*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 1518*x + 1089*x^2.","mathematica":"P[0, x_] := 529 + 1518*x + 1089*x^2;\nP[n_, x_] := P[n, x] = Expand[(23 + 33*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 + 1518*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 33*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 A013610"],"edges":["e-50822203bd5493c4"],"text":"A038337 \u2192 basis A013610"},"headline":"A038337 \u2192 basis A013610","chain":null}