{"id":"n-3bdf31527eaa2fb3","revision":2,"first_index":0,"fingerprint":"f6-3c392e037ce8dc6c","name":"basis_map_A007318 of Pascal-(1,2,1) array read by antidiagonals.","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"],["6","6","1"],["16","24","10","1"],["44","88","58","14","1"],["120","300","272","108","18","1"],["328","984","1136","632","174","22","1"],["896","3136","4400","3160","1232","256","26","1"],["2448","9792","16176","14256","7236","2136","354","30","1"],["6688","30096","57216","59808","37512","14484","3408","468","34","1"],["18272","91360","196464","237696","177816","85224","26280","5112","598","38","1"],["49920","274560","658912","905904","787968","452256","173568","44256","7312","744","42","1"],["136384","818304","2168032","3339040","3312864","2218560","1022400","324864","70300","10072","906","46","1"],["372608","2421952","7021312","11975744","13352512","10230432","5515008","2107776","568600","106556","13456","1084","50","1"],["1017984","7125888","22437248","41986944","51984576","44876224","27742368","12425088","4035304","942456","155424","17528","1278","54","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","51","205","819","3277","13107","52429","209715","838861","3355443","13421773","53687091","214748365"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","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-66e9db1535250539","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["7","2"],["-8","-1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["30","33","4"],["-12","-12","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["12","24","13","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161525,"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-ad7d2c832366f6a0"]},{"id":161526,"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-6f8e74d475c3b6cc","e-ad7d2c832366f6a0"]},{"id":161527,"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-6f8e74d475c3b6cc","e-ad7d2c832366f6a0"]},{"id":161528,"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-6f8e74d475c3b6cc","e-ad7d2c832366f6a0"]},{"id":161529,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-6f8e74d475c3b6cc"]}],"region_exact":{},"incoming_edges":[{"id":"e-ad7d2c832366f6a0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3bdf31527eaa2fb3","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":"9a58031a3ff68f1ccba8c5e9b24d141b1121e8fb15d361f3233ad60a9ab58d31","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-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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-66e9db1535250539","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-3bdf31527eaa2fb3","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":"923420f90001311e6165b5f6514fa484cee6dc2a72394ae8a26cde0bd1860f63","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-6f8e74d475c3b6cc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3bdf31527eaa2fb3","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":"7063c585a92309423c15d8e8405d405c44350d42cdb00311557e590a4df1686b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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":[{"id":"l-1892979835225d7e","kind":"fingerprint_related","node_a_id":"n-3bdf31527eaa2fb3","node_b_id":"n-5397a8ff0da3f463","fingerprint":"f6-3c392e037ce8dc6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3c392e037ce8dc6c"},"other_id":"n-5397a8ff0da3f463","other":{"id":"n-5397a8ff0da3f463","name":"mobius_x_over_x_plus_one of Pascal-(1,2,1) array read by antidiagonals.","first_index":0,"fingerprint":"f6-3c392e037ce8dc6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","10","24","16"]],"row_sums_preview":["1","3","13","51","205","819","3277","13107"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f25cd955a876cde3","steps":["x \u21a6 x/(x+1)"],"edges":["e-5c2436669acea698"],"text":"A081577 \u2192 x \u21a6 x/(x+1)"},"headline":"A081577 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-59a9cb764abac5f3","kind":"oeis_match","node_a_id":"n-3bdf31527eaa2fb3","node_b_id":null,"fingerprint":null,"oeis_id":"A015521","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A015521","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A015521","relation":"match_row_sums"}],"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:    6,     6,     1;\n  3:   16,    24,    10,     1;\n  4:   44,    88,    58,    14,     1;\n  5:  120,   300,   272,   108,    18,     1;\n  6:  328,   984,  1136,   632,   174,    22,    1;\n  7:  896,  3136,  4400,  3160,  1232,   256,   26,   1;\n  8: 2448,  9792, 16176, 14256,  7236,  2136,  354,  30,  1;\n  9: 6688, 30096, 57216, 59808, 37512, 14484, 3408, 468, 34, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (2+x)*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[(2 + 2*x)*P[n - 2, x] + (2 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + (2 + 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-f25cd955a876cde3","steps":["x \u21a6 x+1"],"edges":["e-6f8e74d475c3b6cc"],"text":"A081577 \u2192 x \u21a6 x+1"},"headline":"A081577 \u2192 x \u21a6 x+1","chain":null}