{"id":"n-465696eb3c501fe3","revision":1,"first_index":1,"fingerprint":"f6-f24236d8a2000ecd","name":"basis_map_A008277 of Riordan array (1, 3+x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","3"],["10","28","9"],["33","207","168","27"],["109","1407","2188","837","81"],["360","9216","24624","16884","3753","243"],["1189","59299","257235","282016","108189","15714","729"],["3927","378153","2576100","4248057","2504532","612630","62694","2187"],["12970","2400486","25150363","60090283","51135112","18743409","3175281","241299","6561"],["42837","15202059","241661178","816102813","964068246","496967010","123962616","15397209","903231","19683"],["141481","96154807","2298408354","10783728785","17225097972","12001206484","4135086342","746526537","70879212","3306744","59049"],["467280","607800240","21714325200","139806042912","296364099978","271642880934","124205137956","30569036202","4176844623","312940017","11888532","177147"],["1543321","3840654375","204243707389","1788248814038","4961788847051","5867896493622","3464183350756","1116379581354","205866352872","22022681031","1335097890","42101937","531441"],["5097243","24264623973","1915466105466","22652655387471","81413205690681","122433724811931","91492618693368","37554417432336","8975690909415","1285585734339","110569987332","5534977698","147209157","1594323"],["16835050","153285887638","17928560597655","284935501929591","1315753244571921","2488215831495584","2318763346945435","1188179094457756","357976083608415","65920737921597","7542275542794","532809480258","22395100887","509120478","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","47","435","4622","55080","724371","10388280","160955764","2674326882","47360595767","889411641021","17635849757077","367863888254733","8045923698278028"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162306,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-38e3f340ba87c13c"]},{"id":162307,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-38e3f340ba87c13c"]},{"id":162308,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-38e3f340ba87c13c"]}],"region_exact":{},"incoming_edges":[{"id":"e-38e3f340ba87c13c","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-465696eb3c501fe3","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":"b959c3aea85e879a95ee2b1a2afdddd08bd6e3d63300bf06ba29f4c3b0bd088d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5d39164c4108e043","role":"sequence","name":"Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,        3;\n   3:    10,       28,         9;\n   4:    33,      207,       168,        27;\n   5:   109,     1407,      2188,       837,        81;\n   6:   360,     9216,     24624,     16884,      3753,       243;\n   7:  1189,    59299,    257235,    282016,    108189,     15714,       729;\n   8:  3927,   378153,   2576100,   4248057,   2504532,    612630,     62694,     2187;\n   9: 12970,  2400486,  25150363,  60090283,  51135112,  18743409,   3175281,   241299,   6561;\n  10: 42837, 15202059, 241661178, 816102813, 964068246, 496967010, 123962616, 15397209, 903231, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + x*d/dx P(n-2,x) + (3+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (3 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 3*x; for(n = 3, N, my(i = n); v[i] = (1 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (3 + 3*x)*v[i - 1] + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5d39164c4108e043","steps":["basis A008277"],"edges":["e-38e3f340ba87c13c"],"text":"A099097 \u2192 basis A008277"},"headline":"A099097 \u2192 basis A008277","chain":null}
