{"id":"n-913df248d683398b","revision":0,"first_index":1,"fingerprint":"f6-fb43674541c36216","name":"shift_x_plus_one of basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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"],["13","9"],["121","150","45"],["1093","1911","1071","189"],["9841","22140","17982","6156","729"],["88573","243573","258786","131274","31185","2673"],["797161","2590770","3400371","2287116","820287","145314","9477"],["7174453","26904195","42037785","35241615","16952895","4608009","637875","32805"],["64570081","274422840","497391372","499535208","301824630","111140424","23943276","2676888","111537"],["581130733","2760370977","5693265108","6663707892","4850192358","2258553294","665014212","117179460","10845333","373977"],["5230176601","27458427150","63497612745","84878002440","72299593170","40757763060","15269023098","3706545096","546852789","42712110","1240029"],["47071589413","270661639119","693570450195","1042470219105","1017426087090","673771376502","306945990918","95327738946","19515832281","2455040907","164333367","4074381"],["423644304721","2647776904500","7446872543850","12432139996500","13679103288375","10410155721000","5590330886700","2119932145800","558064061775","98058737700","10671925770","620014500","13286025"],["3812798742493","25736391511821","78817699004886","144700180400574","177220440083295","152416333012167","94317009713892","42294386902644","13646177782731","3097641747915","473759456742","45141661422","2300608089","43046721"],["34315188682441","248785117947690","824100703201647","1650145040136756","2226493180809369","2135681449973190","1496860384658247","775449180547992","296733766082499","82846413112086","16439763888405","2213769921012","186528882267","8414836794","138706101"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","22","316","4264","56848","756064","10050496","133589632","1775616256","23600633344","313687948288","4169384372224","55417383817216","736580303675392","9790259041386496"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-39","-27"],["26","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-16","-28","-12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-4e7c1404f4ed504f","recurrence":{"first_index":1,"initial_polynomials":[["1"],["13","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","-18","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185554,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-bbb35446d7d55c61"]},{"id":185555,"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-79bfad215438006d","e-bbb35446d7d55c61"]},{"id":185556,"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-79bfad215438006d","e-bbb35446d7d55c61"]},{"id":185557,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-79bfad215438006d","e-bbb35446d7d55c61"]},{"id":185558,"position":4,"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-79bfad215438006d"]}],"region_exact":{},"incoming_edges":[{"id":"e-79bfad215438006d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-913df248d683398b","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"cbff866f0e556e14b409cc8b285b26259ecadd933b70b4c77ba8ce72473d61e1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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]"}]},{"id":"e-4e7c1404f4ed504f","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-913df248d683398b","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"2942a5466b2aeb6d3c6c71b39c249c2b3279e7ac0d1bd25197fc5ad818323152","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-bbb35446d7d55c61","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-913df248d683398b","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"adaaf536d0102b29ffc3b3ae7acafd96b25c8ca85e34d3e4c95db668e754c9f0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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:        13,          9;\n   3:       121,        150,         45;\n   4:      1093,       1911,       1071,        189;\n   5:      9841,      22140,      17982,       6156,        729;\n   6:     88573,     243573,     258786,     131274,      31185,       2673;\n   7:    797161,    2590770,    3400371,    2287116,     820287,     145314,      9477;\n   8:   7174453,   26904195,   42037785,   35241615,   16952895,    4608009,    637875,     32805;\n   9:  64570081,  274422840,  497391372,  499535208,  301824630,  111140424,  23943276,   2676888,   111537;\n  10: 581130733, 2760370977, 5693265108, 6663707892, 4850192358, 2258553294, 665014212, 117179460, 10845333, 373977;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = (26*n-39+(18*n-27)*x)*P(n-1,x) - (16+28*x+12*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((26*n - 39 + (18*n - 27)*x)*P[n - 1, x] - (16 + 28*x + 12*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((26*n - 39 + (18*n - 27)*x)*v[i - 1] - (16 + 28*x + 12*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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-ba8de4e02acfb6ca","steps":["basis A013610","basis A007318"],"edges":["e-4aa0aeb00b1923c9","e-79bfad215438006d"],"text":"A091042 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A091042 \u2192 basis A013610 \u2192 basis A007318","chain":null}
