{"id":"n-c5c52c7bc39f20ea","revision":1,"first_index":0,"fingerprint":"f6-ea09498ac2aae330","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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"],["9","1"],["81","19","1"],["729","271","30","1"],["6561","3439","601","42","1"],["59049","40951","10050","1105","55","1"],["531441","468559","151501","23310","1820","69","1"],["4782969","5217031","2135070","431221","46970","2786","84","1"],["43046721","56953279","28702801","7309722","1041831","85974","4046","100","1"],["387420489","612579511","372684090","116419465","20853525","2245467","146664","5646","117","1"],["3486784401","6513215599","4712104501","1769717670","387515290","52290063","4445427","237000","7635","135","1"],["31381059609","68618940391","58346365110","25948716541","6807416440","1119576172","118971468","8237427","366795","10065","154","1"],["282429536481","717570463519","710428956601","369730963602","114445130261","22481482848","2904148192","250770300","14472942","547965","12991","174","1"],["2541865828329","7458134171671","8532288986130","5147200519825","1857517656995","429185890133","66043705728","6916472992","496810314","24336312","794794","16471","195","1"],["22876792454961","77123207545039","101313313019101","70298695224030","29294930060760","7866120118857","1419841476053","176707273600","15362248330","934863930","39437398","1124214","20566","217","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","101","1031","10644","111211","1176701","12616132","137144475","1512354975","16926317722","192349660173","2220256485877","26039675189890","310385944867057"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["9","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":166052,"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-6eda6604dc872938"]},{"id":166053,"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-6eda6604dc872938"]},{"id":166054,"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-6eda6604dc872938"]}],"region_exact":{},"incoming_edges":[{"id":"e-6eda6604dc872938","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-c5c52c7bc39f20ea","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":"af83cb1b15519f0ac3c8f6e920b9dc753fe0c43405eadc6531c8a0c8dd324369","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-366b2525683554ec","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^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:         1;\n  1:         9,         1;\n  2:        81,        19,         1;\n  3:       729,       271,        30,         1;\n  4:      6561,      3439,       601,        42,        1;\n  5:     59049,     40951,     10050,      1105,       55,       1;\n  6:    531441,    468559,    151501,     23310,     1820,      69,      1;\n  7:   4782969,   5217031,   2135070,    431221,    46970,    2786,     84,    1;\n  8:  43046721,  56953279,  28702801,   7309722,  1041831,   85974,   4046,  100,   1;\n  9: 387420489, 612579511, 372684090, 116419465, 20853525, 2245467, 146664, 5646, 117, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(9 + 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; for(n = 1, N, my(i = n + 1); v[i] = (9 + 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-366b2525683554ec","steps":["basis A008277"],"edges":["e-6eda6604dc872938"],"text":"A038279 \u2192 basis A008277"},"headline":"A038279 \u2192 basis A008277","chain":null}
