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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-a91be9e7e71528cb"]},{"id":201044,"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-a91be9e7e71528cb"]},{"id":201045,"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 real-rootedness)","region_exact":null,"edges":["e-a91be9e7e71528cb"]}],"region_exact":{},"incoming_edges":[{"id":"e-a91be9e7e71528cb","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-25f8fdf7d4ce854d","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"0ae24cf7540fa01c7cd80296c4d72dca7d04e4305302ebdcce67df79173cd4f4","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-2ff1c5d05cc7026f","role":"sequence","name":"basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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:              45,               114,                 72;\n  2:            2025,             10260,              19476,               16416,                5184;\n  3:           91125,            692550,            2191860,             3697704,             3506976,              1772928,               373248;\n  4:         4100625,          41553000,          184145400,           466132320,           737166096,            745811712,            471412224,             170201088,              26873856;\n  5:       184528125,        2337356250,        13318830000,         44960346000,         99570027600,         151159989024,         159312044160,          115098485760,           54553927680,           15318097920,            1934917632;\n  6:      8303765625,      126217237500,       879091987500,       3709851840000,      10565086446000,       21390327564480,       31570322723136,        34224524103168,        27046621301760,        15195553136640,         5761217249280,         1323483660288,          139314069504;\n  7:    373669453125,     6626404968750,     54545775637500,     276247460475000,     961646622930000,     2434093927725600,     4619848089003840,      6679223959722624,      7391756942406144,      6231280454977536,      3938904567521280,      1810415356968960,       571953912348672,       111172627464192,       10030613004288;\n  8:  16815125390625,   340786541250000,   3236874270750000,   19126455301800000,   78693604371900000,   239051758915872000,   554618428616851200,   1002482523130199040,   1426689656224932096,   1603972037008318464,   1419823177259139072,    979156004519411712,    515726405611683840,    200555419945402368,    54305738805215232,     9147919059910656,     722204136308736;\n  9: 756680642578125, 17252318650781250, 185719696914375000, 1254230786416500000, 5954683048634700000, 21105504831340440000, 57875669318944512000, 125549941045122777600, 218416569027377921280, 307000104140390920704, 349466510443804674048, 321407849075514310656, 237058741530396721152, 138317036462672707584, 62439377324051791872, 21042500817559486464, 4985375152939204608, 740981443852763136, 51998697814228992;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (45+114*x+72*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(45 + 114*x + 72*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; for(n = 1, N, my(i = n + 1); v[i] = (45 + 114*x + 72*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-eed4f744ce063213","steps":["basis A013610","basis A013609"],"edges":["e-bf76d3591eaf645b","e-a91be9e7e71528cb"],"text":"A200536 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A200536 \u2192 basis A013610 \u2192 basis A013609","chain":null}