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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-8c6da351df327041"]},{"id":198098,"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-8c6da351df327041"]},{"id":198099,"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-8c6da351df327041"]}],"region_exact":{},"incoming_edges":[{"id":"e-8c6da351df327041","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-395eb038698327b7","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":"a997922819614adc0c81c027925d54f2e451eef315d0a7fd689ce23eee393c81","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-dba956a9ceeed98f","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","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:             441,              235,               25;\n  1:            9261,             8315,             1950,               125;\n  2:          194481,           262495,           102025,             14250,              625;\n  3:         4084101,          7797275,          4475250,           1023125,            96875,             3125;\n  4:        85766121,        223149655,        177719125,          59208750,          9087500,           628125,           15625;\n  5:      1801088541,       6230721635,       6625041150,        3020110625,        668631250,         74331250,         3937500,          78125;\n  6:     37822859361,     171004205215,     236529883825,      141849188250,      42514434375,       6762393750,       572468750,       24062500,        390625;\n  7:    794280046581,    4635223632395,    8187447424650,     6289220196125,    2452337750625,     523642284375,     63007875000,     4209843750,     144140625,      1953125;\n  8:  16679880978201,  124487214675175,  276986988326125,   267349164183750,  131991948756250,   36349233834375,   5831613046875,   550790625000,   29841796875,    849609375,    9765625;\n  9: 350277500542221, 3320066986445555, 9209032711485750, 11009504852245625, 6748415719925000, 2332024500162500, 479158434562500, 60002340234375, 4574302734375, 205283203125, 4941406250, 48828125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (21+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 441 + 235*x + 25*x^2.","mathematica":"P[0, x_] := 441 + 235*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(21 + 5*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 441 + 235*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 5*x)*v[i - 1] + 5*x*deriv(v[i - 1], 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-2bc1923905556bf6","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-2406b2ec50933490","e-8c6da351df327041"],"text":"A038319 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A038319 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}