{"id":"n-f623dd49245e9923","revision":1,"first_index":0,"fingerprint":"f6-b123934b80f387c2","name":"mobius_x_over_x_plus_one 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"],["1","2"],["1","9","9"],["1","16","42","28"],["1","23","124","202","101"],["1","30","255","740","855","342"],["1","37","435","1985","3965","3567","1189"],["1","44","664","4280","12910","19472","14308","4088"],["1","51","942","7968","33570","75042","90912","56484","14121"],["1","58","1269","13392","74226","228132","402906","407208","219069","48682"],["1","65","1645","20895","145560","584274","1402230","2042790","1770465","839845","167969"],["1","72","2070","30820","260655","1313484","4082568","8002440","9906615","7512160","3186414","579348"],["1","79","2544","43510","434995","2667069","10358706","26100096","43136595","46391665","31262284","11991246","1998541"],["1","86","3067","59308","686465","4994434","23563539","73797768","155761827","222210130","211139489","128008684","44809843","6893822"],["1","93","3639","78557","1035351","8759889","49062727","186053721","485651373","879917239","1103201049","938557407","517094861","166462443","23780349"]],"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:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["12","108","360","600","540","252","48"],["-18","-155","-513","-861","-785","-372","-72"],["7","52","162","268","247","120","24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","65","406","1090","1565","1261","540","96"],["-1","-30","-182","-496","-729","-602","-264","-48"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","29","154","370","485","361","144","24"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164274,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-28cdd807988aaaf4"]},{"id":164275,"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-28cdd807988aaaf4"]},{"id":164276,"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-28cdd807988aaaf4"]}],"region_exact":{},"incoming_edges":[{"id":"e-28cdd807988aaaf4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-f623dd49245e9923","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"5dfb756b0785a4dfdae16c46922b44a84332bfc0265a0e7e801c615441e016c3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"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-4f8c7533ab8b37bb","other":{"id":"n-4f8c7533ab8b37bb","name":"basis_map_A007318 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"],["2","1"],["9","9","1"]],"card_rows":[["1"],["2","1"],["9","9","1"],["28","42","16","1"]],"row_sums_preview":["1","3","19","87","451","2223","11179","55767"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6ad01dc6e705cdb6","steps":["basis A007318"],"edges":["e-1ae1e4b2be227437"],"text":"A081580 \u2192 basis A007318"},"headline":"A081580 \u2192 basis A007318","chain":null}},{"id":"l-d02e74e2e67a42aa","kind":"oeis_match","node_a_id":"n-f623dd49245e9923","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: 1,  2;\n  2: 1,  9,    9;\n  3: 1, 16,   42,    28;\n  4: 1, 23,  124,   202,   101;\n  5: 1, 30,  255,   740,   855,    342;\n  6: 1, 37,  435,  1985,  3965,   3567,   1189;\n  7: 1, 44,  664,  4280, 12910,  19472,  14308,   4088;\n  8: 1, 51,  942,  7968, 33570,  75042,  90912,  56484,  14121;\n  9: 1, 58, 1269, 13392, 74226, 228132, 402906, 407208, 219069, 48682;\n","recurrence_sentence":"The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (7*n^2-18*n+12+(52*n^2-155*n+108)*x+(162*n^2-513*n+360)*x^2+(268*n^2-861*n+600)*x^3+(247*n^2-785*n+540)*x^4+(120*n^2-372*n+252)*x^5+(24*n^2-72*n+48)*x^6)*P(n-1,x) - (n-1+(30*n-65)*x+(182*n-406)*x^2+(496*n-1090)*x^3+(729*n-1565)*x^4+(602*n-1261)*x^5+(264*n-540)*x^6+(48*n-96)*x^7)*d/dx P(n-1,x) + (x+29*x^2+154*x^3+370*x^4+485*x^5+361*x^6+144*x^7+24*x^8)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n^2 - 18*n + 12 + (52*n^2 - 155*n + 108)*x + (162*n^2 - 513*n + 360)*x^2 + (268*n^2 - 861*n + 600)*x^3 + (247*n^2 - 785*n + 540)*x^4 + (120*n^2 - 372*n + 252)*x^5 + (24*n^2 - 72*n + 48)*x^6)*P[n - 1, x] - (n - 1 + (30*n - 65)*x + (182*n - 406)*x^2 + (496*n - 1090)*x^3 + (729*n - 1565)*x^4 + (602*n - 1261)*x^5 + (264*n - 540)*x^6 + (48*n - 96)*x^7)*D[P[n - 1, x], x] + (x + 29*x^2 + 154*x^3 + 370*x^4 + 485*x^5 + 361*x^6 + 144*x^7 + 24*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^2 - 18*n + 12 + (52*n^2 - 155*n + 108)*x + (162*n^2 - 513*n + 360)*x^2 + (268*n^2 - 861*n + 600)*x^3 + (247*n^2 - 785*n + 540)*x^4 + (120*n^2 - 372*n + 252)*x^5 + (24*n^2 - 72*n + 48)*x^6)*v[i - 1] - (n - 1 + (30*n - 65)*x + (182*n - 406)*x^2 + (496*n - 1090)*x^3 + (729*n - 1565)*x^4 + (602*n - 1261)*x^5 + (264*n - 540)*x^6 + (48*n - 96)*x^7)*deriv(v[i - 1], x) + (x + 29*x^2 + 154*x^3 + 370*x^4 + 485*x^5 + 361*x^6 + 144*x^7 + 24*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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":["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}
