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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-a91ee3c8b8082a44"]},{"id":197936,"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-a91ee3c8b8082a44"]},{"id":197937,"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-a91ee3c8b8082a44"]}],"region_exact":{},"incoming_edges":[{"id":"e-a91ee3c8b8082a44","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-269219ae86b544b7","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7bece481215c82ebbe7f383764f79348b445cec46ac92f15c8e72341bc79d72c","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-83675781d9687218","role":"sequence","name":"basis_map_A013609 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:             28,                38,                   8;\n  2:            784,              3572,                3236,                  864,                    64;\n  3:          21952,            265544,              587480,               419256,                117856,                 13440,                    512;\n  4:         614656,          18360080,            85012592,            122906592,              73284112,              20455680,                2782720,                 176128,                  4096;\n  5:       17210368,        1235122208,         11197872608,          29450014560,           31476482560,           16190479200,             4380272256,              647490560,              51875840,               2088960,                32768;\n  6:      481890304,       82172059712,       1410578995136,        6389234826624,        11309793419584,         9566202475200,          4334903445056,          1124399368704,          172628917248,           15767470080,            832159744,            23199744,             262144;\n  7:    13492928512,     5441667772544,     173710625763200,     1313352523195776,      3682261810855936,      4820737343036160,       3369402348178048,       1365778283722624,       337989688319488,        52567621908480,        5189387083776,        321670250496,        12025724928,         245891072,          2097152;\n  8:   377801998336,   359662804271360,   21139233094729472,   261740256552801792,   1130642261797180672,   2208336240560540160,    2269528665661426176,    1350506132145400832,    495453918233018624,    116761484856238080,    18124853324398592,    1874574370013184,    128872146665472,     5774505934848,     160922861568,     2516582400,      16777216;\n  9: 10578455953408, 23752101557846528, 2556132785214176768, 51213272364097574400, 334736040184320655360, 952170531687619023360, 1395315032648588567552, 1172602394374717953024, 607186205368856060928, 203505764157845521920, 45659056139621091328, 7008498451718307840, 744620847642836992, 54858064197844992, 2775926122217472, 94090611916800, 2029875363840, 25065160704, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (28+38*x+8*x^2)*P(n-1,x) + (38*x+16*x^2)*d/dx P(n-1,x) + 8*x^2*d^2/dx^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[(28 + 38*x + 8*x^2)*P[n - 1, x] + (38*x + 16*x^2)*D[P[n - 1, x], x] + 8*x^2*D[P[n - 1, x], {x, 2}]];\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] = (28 + 38*x + 8*x^2)*v[i - 1] + (38*x + 16*x^2)*deriv(v[i - 1], x) + 8*x^2*deriv(deriv(v[i - 1], x), 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-eed4f744ce063213","steps":["basis A013609","basis A008277"],"edges":["e-185113cbc4b6c77b","e-a91ee3c8b8082a44"],"text":"A200536 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A200536 \u2192 basis A013609 \u2192 basis A008277","chain":null}