{"id":"n-4f8c7533ab8b37bb","revision":2,"first_index":0,"fingerprint":"f6-b123934b80f387c2","name":"basis_map_A007318 of Pascal-(1,5,1) array.","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"],["9","9","1"],["28","42","16","1"],["101","202","124","23","1"],["342","855","740","255","30","1"],["1189","3567","3965","1985","435","37","1"],["4088","14308","19472","12910","4280","664","44","1"],["14121","56484","90912","75042","33570","7968","942","51","1"],["48682","219069","407208","402906","228132","74226","13392","1269","58","1"],["167969","839845","1770465","2042790","1402230","584274","145560","20895","1645","65","1"],["579348","3186414","7512160","9906615","8002440","4082568","1313484","260655","30820","2070","72","1"],["1998541","11991246","31262284","46391665","43136595","26100096","10358706","2667069","434995","43510","2544","79","1"],["6893822","44809843","128008684","211139489","222210130","155761827","73797768","23563539","4994434","686465","59308","3067","86","1"],["23780349","166462443","517094861","938557407","1103201049","879917239","485651373","186053721","49062727","8759889","1035351","78557","3639","93","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","19","87","451","2223","11179","55767","279091","1394943","6975739","34876647","174387331","871928463","4359658699"],"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":[["5","5"]]},"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-964c28f05df17150","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["13","2"],["-17","-1"],["6","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["60","63","4"],["-24","-24","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["24","48","25","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161624,"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-1ae1e4b2be227437"]},{"id":161625,"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-1ae1e4b2be227437","e-dd36dc372b4d6491"]},{"id":161626,"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-1ae1e4b2be227437","e-dd36dc372b4d6491"]},{"id":161627,"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-1ae1e4b2be227437","e-dd36dc372b4d6491"]},{"id":161628,"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-dd36dc372b4d6491"]}],"region_exact":{},"incoming_edges":[{"id":"e-1ae1e4b2be227437","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4f8c7533ab8b37bb","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":"afdad1850c3ebfd313833a7316aeac66c207bb418f0215f53eebdc51455bd9e0","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-6ad01dc6e705cdb6","role":"sequence","name":"Pascal-(1,5,1) array.","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-964c28f05df17150","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-4f8c7533ab8b37bb","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":"99cfd5ade0bc0cb3cf66c9053979278ba48d8986a05bd3c2b3c29e57465002c1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-dd36dc372b4d6491","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-4f8c7533ab8b37bb","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":"3fd040531d01bb8bf9b472950e9bf5a67875776d0c7ccc6268049709c3a67c07","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-6ad01dc6e705cdb6","role":"sequence","name":"Pascal-(1,5,1) array.","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-1ac9a598972398ae","kind":"fingerprint_related","node_a_id":"n-4f8c7533ab8b37bb","node_b_id":"n-f623dd49245e9923","fingerprint":"f6-b123934b80f387c2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b123934b80f387c2"},"other_id":"n-f623dd49245e9923","other":{"id":"n-f623dd49245e9923","name":"mobius_x_over_x_plus_one of Pascal-(1,5,1) array.","first_index":0,"fingerprint":"f6-b123934b80f387c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","9","9"]],"card_rows":[["1"],["1","2"],["1","9","9"],["1","16","42","28"]],"row_sums_preview":["1","3","19","87","451","2223","11179","55767"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6ad01dc6e705cdb6","steps":["x \u21a6 x/(x+1)"],"edges":["e-28cdd807988aaaf4"],"text":"A081580 \u2192 x \u21a6 x/(x+1)"},"headline":"A081580 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f24daccba7b05da3","kind":"oeis_match","node_a_id":"n-4f8c7533ab8b37bb","node_b_id":null,"fingerprint":null,"oeis_id":"A015528","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":"A015528","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A015528","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:     9,      9,      1;\n  3:    28,     42,     16,      1;\n  4:   101,    202,    124,     23,      1;\n  5:   342,    855,    740,    255,     30,     1;\n  6:  1189,   3567,   3965,   1985,    435,    37,     1;\n  7:  4088,  14308,  19472,  12910,   4280,   664,    44,    1;\n  8: 14121,  56484,  90912,  75042,  33570,  7968,   942,   51,  1;\n  9: 48682, 219069, 407208, 402906, 228132, 74226, 13392, 1269, 58, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+5*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[(5 + 5*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] = (5 + 5*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-6ad01dc6e705cdb6","steps":["basis A007318"],"edges":["e-1ae1e4b2be227437"],"text":"A081580 \u2192 basis A007318"},"headline":"A081580 \u2192 basis A007318","chain":null}
