{"id":"n-317113238d23620b","revision":2,"first_index":0,"fingerprint":"f6-1cfe01b94310c21a","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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"],["5","1"],["22","10","1"],["95","69","15","1"],["409","410","141","20","1"],["1760","2252","1070","238","25","1"],["7573","11790","7179","2200","360","30","1"],["32585","59767","44475","17465","3925","507","35","1"],["140206","296050","260605","125200","36010","6370","679","40","1"],["603275","1441155","1465650","834210","293475","66339","9660","876","45","1"],["2595757","6920900","7987590","5261100","2193555","606060","112602","13920","1098","50","1"],["11168960","32876792","42461900","31790460","15348450","5024838","1140090","179574","19275","1345","55","1"],["48057529","154790220","221223522","185630900","101952045","38654460","10387482","1996200","272655","25850","1617","60","1"],["206780765","723378253","1133522130","1054006642","649345775","280149831","87171600","19829760","3301650","397870","33770","1914","65","1"],["889731238","3359301370","5727318337","5846662640","3994879382","1934131550","684845385","180331800","35520045","5213450","561869","43160","2236","70","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","33","180","981","5346","29133","158760","865161","4714686","25692633","140011740","762992541","4157920026","22658542533"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","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-7c7349737e12ea76","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["5","1"],["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["13","10","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161789,"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-c60a744f47ece880"]},{"id":161790,"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-c60a744f47ece880","e-f1629c32ad3d0109"]},{"id":161791,"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-c60a744f47ece880","e-f1629c32ad3d0109"]},{"id":161792,"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-c60a744f47ece880","e-f1629c32ad3d0109"]},{"id":161793,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-9, -1)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-9"},"upper":{"closed":false,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-9, -1)","region_exact":null,"edges":["e-f1629c32ad3d0109"]}],"region_exact":{},"incoming_edges":[{"id":"e-c60a744f47ece880","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-317113238d23620b","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":"178fd98a22573270f494dac83212abd759bc50f84f3cec3ee6f3bb2a644cfb2a","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-6466a8550c08a52f","role":"sequence","name":"Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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-7c7349737e12ea76","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-317113238d23620b","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":"4c860f7edb8d731bde1ace49850b663a4071db2ff289c422385d6f99fbc3ebc1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-f1629c32ad3d0109","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-317113238d23620b","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":"450f8c9a8d29ae3f229f0cde5ceab149ff6c0c0c67b8e9099ad3c88eac7604b8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-9"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6466a8550c08a52f","role":"sequence","name":"Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1a5cc811e9993e73","kind":"oeis_match","node_a_id":"n-317113238d23620b","node_b_id":null,"fingerprint":null,"oeis_id":"A138395","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A138395","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A138395","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-317113238d23620b","oeis_id":"A138395","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":15655}],"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:      5,       1;\n  2:     22,      10,       1;\n  3:     95,      69,      15,      1;\n  4:    409,     410,     141,     20,      1;\n  5:   1760,    2252,    1070,    238,     25,     1;\n  6:   7573,   11790,    7179,   2200,    360,    30,    1;\n  7:  32585,   59767,   44475,  17465,   3925,   507,   35,   1;\n  8: 140206,  296050,  260605, 125200,  36010,  6370,  679,  40,  1;\n  9: 603275, 1441155, 1465650, 834210, 293475, 66339, 9660, 876, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -3*P(n-2,x) + (5+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + x;\nP[n_, x_] := P[n, x] = Expand[-3*P[n - 2, x] + (5 + 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] = 5 + x; for(n = 2, N, my(i = n + 1); v[i] = -3*v[i - 2] + (5 + 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-6466a8550c08a52f","steps":["basis A007318"],"edges":["e-c60a744f47ece880"],"text":"A116414 \u2192 basis A007318"},"headline":"A116414 \u2192 basis A007318","chain":null}