{"id":"n-c2648604ab227c74","revision":2,"first_index":0,"fingerprint":"f6-2b83e2879aa49b31","name":"basis_map_A007318 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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","2"],["8","9","2"],["21","33","15","2"],["55","111","76","21","2"],["144","355","324","137","27","2"],["377","1098","1251","714","216","33","2"],["987","3316","4527","3256","1335","313","39","2"],["2584","9837","15646","13581","7045","2241","428","45","2"],["6765","28779","52248","53133","33381","13455","3486","561","51","2"],["17711","83265","169877","198066","146231","71505","23485","5124","712","57","2"],["46368","238727","540648","710942","603378","347291","138474","38296","7209","881","63","2"],["121393","679284","1690794","2475408","2374845","1573746","739228","248238","59211","9795","1068","69","2"],["317811","1920518","5211018","8406076","8996565","6748792","3652956","1445646","418662","87715","12936","1273","75","2"],["832040","5400081","15862778","27953838","33020926","27669195","16968432","7741656","2642421","672012","125455","16686","1496","81","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","19","71","265","989","3691","13775","51409","191861","716035","2672279","9973081","37220045","138907099"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":"e-11e03be131215839","recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["8"],["-16"],["8"]]},"terms":[{"coeff":{"coeffs":[["18","4"],["-27","-8"],["9","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-8","-18","-4"],["20","24","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["15","18","3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161506,"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-06cd9a9d948af774"]},{"id":161507,"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-06cd9a9d948af774","e-66d94dddc72ac30f"]},{"id":161508,"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-06cd9a9d948af774","e-66d94dddc72ac30f"]},{"id":161509,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-06cd9a9d948af774","e-66d94dddc72ac30f"]},{"id":161510,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-5, -1)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)","region_exact":null,"edges":["e-66d94dddc72ac30f"]}],"region_exact":{},"incoming_edges":[{"id":"e-06cd9a9d948af774","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c2648604ab227c74","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":"3158f173bb1039a7cee92c38f997eaa70f6a022b38be2128ddf52d45613ea35c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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]"}]},{"id":"e-11e03be131215839","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-c2648604ab227c74","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":"42e184a828a143797c874d0ebd8daa55386359bffd54efa5674954fdae677fa0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-66d94dddc72ac30f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c2648604ab227c74","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":"41816ba76f5fb24775c211ee05a7632827b9e19bd00e88a0b1c681d4cd2ed41c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-161dece809114fc8","kind":"oeis_match","node_a_id":"n-c2648604ab227c74","node_b_id":null,"fingerprint":null,"oeis_id":"A001834","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001834","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001834","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-c2648604ab227c74","oeis_id":"A001834","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":12402}],"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,     2;\n  2:    8,     9,     2;\n  3:   21,    33,    15,     2;\n  4:   55,   111,    76,    21,     2;\n  5:  144,   355,   324,   137,    27,     2;\n  6:  377,  1098,  1251,   714,   216,    33,    2;\n  7:  987,  3316,  4527,  3256,  1335,   313,   39,   2;\n  8: 2584,  9837, 15646, 13581,  7045,  2241,  428,  45,  2;\n  9: 6765, 28779, 52248, 53133, 33381, 13455, 3486, 561, 51, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + x)*P[n - 1, 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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + x)*v[i - 1]); 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-353fb5a029e4e669","steps":["basis A007318"],"edges":["e-06cd9a9d948af774"],"text":"A207607 \u2192 basis A007318"},"headline":"A207607 \u2192 basis A007318","chain":null}