{"id":"n-07e5e6cd238b2e3e","revision":1,"first_index":0,"fingerprint":"f6-b3fe52a70c901152","name":"basis_map_A008277 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["10","8","1"],["33","47","14","1"],["109","247","128","21","1"],["360","1224","976","272","29","1"],["1189","5859","6735","2820","501","38","1"],["3927","27433","43686","25719","6744","840","48","1"],["12970","126566","271927","216015","78252","14193","1317","59","1"],["42837","578027","1644692","1714579","823218","203580","27222","1963","71","1"],["141481","2621047","9743834","13068153","8084380","2615268","471990","48641","2812","84","1"],["467280","11824560","56852320","96655760","75489482","31051222","7257312","1001326","82175","3901","98","1"],["1543321","53149095","327938709","698803326","678572187","347675246","102286228","18131690","1979200","132639","5270","113","1"],["5097243","238256101","1875224160","4964234025","5922911805","3721770969","1350103704","298871048","41643791","3692053","206128","6962","129","1"],["16835050","1065963158","10651342123","34783780323","50516280921","38461703220","16936383147","4586153932","792897239","89254017","6562326","310222","9023","146","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","19","95","506","2862","17143","108398","721300","5036190","36797691","280685437","2230217025","18422018119","157907474848"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"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","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162294,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 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-781e4104e926c42c"]},{"id":162295,"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-781e4104e926c42c"]},{"id":162296,"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-781e4104e926c42c"]}],"region_exact":{},"incoming_edges":[{"id":"e-781e4104e926c42c","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-07e5e6cd238b2e3e","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":"f76d729ad44a73dacfffc4a37a54283beecaf05a7c48089a85021cda284b3e31","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-d5c14b9fec875ecc","role":"sequence","name":"Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:     3,      1;\n  2:    10,      8,       1;\n  3:    33,     47,      14,       1;\n  4:   109,    247,     128,      21,      1;\n  5:   360,   1224,     976,     272,     29,      1;\n  6:  1189,   5859,    6735,    2820,    501,     38,     1;\n  7:  3927,  27433,   43686,   25719,   6744,    840,    48,    1;\n  8: 12970, 126566,  271927,  216015,  78252,  14193,  1317,   59,  1;\n  9: 42837, 578027, 1644692, 1714579, 823218, 203580, 27222, 1963, 71, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + x*d/dx P(n-2,x) + (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (3 + x)*P[n - 1, x] + 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] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (3 + x)*v[i - 1] + 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-d5c14b9fec875ecc","steps":["basis A008277"],"edges":["e-781e4104e926c42c"],"text":"A121574 \u2192 basis A008277"},"headline":"A121574 \u2192 basis A008277","chain":null}
