{"id":"n-eca894b6614d2b87","revision":1,"first_index":0,"fingerprint":"f6-4b4db2101844ae02","name":"basis_map_A008277 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","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"],["2","1"],["9","10","1"],["40","83","25","1"],["251","840","430","49","1"],["1662","8745","7142","1516","84","1"],["13615","107068","124643","40388","4234","132","1"],["118640","1388179","2265045","1052913","172272","10108","195","1"],["1203945","20453498","44599146","27849455","6457163","599874","21537","275","1"],["12966850","320219261","927165028","760272454","236962730","31386161","1793267","42079","374","1"],["157163705","5563717202","20842231397","21732196592","8736793264","1552814220","127551294","4759311","76775","494","1"],["2012984760","102444419043","495257221985","649344598921","328959235800","75318132980","8329237502","449401862","11480356","132513","637","1"],["28386343795","2058458948784","12637487072898","20446487970165","12762605265081","3654506347180","521031391086","38045850470","1408999752","25603824","218432","805","1"],["421122929150","43652993125273","340075518625014","674636973721812","512261600364352","179681263791421","32008483460282","3021769929354","152179084936","4008434782","53478128","346366","1000","1"],["6774991711455","997748688749836","9757877382628231","23447759128527696","21358445063682054","9020165906669948","1961809883193115","231712635547282","15145472060456","544264807866","10503546702","105652114","531328","1224","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","20","149","1571","19150","290081","5007353","101184894","2290808205","58717304255","1662126846360","52148444012273","1785915967291871","66797994027309308"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","1"],["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","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":166007,"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-f0df87e08c6880bc"]},{"id":166008,"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-f0df87e08c6880bc"]},{"id":166009,"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-f0df87e08c6880bc"]}],"region_exact":{},"incoming_edges":[{"id":"e-f0df87e08c6880bc","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-eca894b6614d2b87","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":"f94e083282033440a7c4cff866660d1503bfdcf1bbb837b4b07d45e4401b0474","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-74f9a0f107d622a7","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","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:        2,         1;\n  2:        9,        10,         1;\n  3:       40,        83,        25,         1;\n  4:      251,       840,       430,        49,         1;\n  5:     1662,      8745,      7142,      1516,        84,        1;\n  6:    13615,    107068,    124643,     40388,      4234,      132,       1;\n  7:   118640,   1388179,   2265045,   1052913,    172272,    10108,     195,     1;\n  8:  1203945,  20453498,  44599146,  27849455,   6457163,   599874,   21537,   275,   1;\n  9: 12966850, 320219261, 927165028, 760272454, 236962730, 31386161, 1793267, 42079, 374, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2+n-1+(n^2+n-1)*x)*P(n-2,x) + (n^2+n-1)*x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(n^2 + n - 1 + (n^2 + n - 1)*x)*P[n - 2, x] + (n^2 + n - 1)*x*D[P[n - 2, x], x] + (2 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (n^2 + n - 1 + (n^2 + n - 1)*x)*v[i - 2] + (n^2 + n - 1)*x*deriv(v[i - 2], x) + (2 + 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-74f9a0f107d622a7","steps":["basis A008277"],"edges":["e-f0df87e08c6880bc"],"text":"A154233 \u2192 basis A008277"},"headline":"A154233 \u2192 basis A008277","chain":null}