{"id":"n-26fdd4314504ef0a","revision":0,"first_index":0,"fingerprint":"f6-063dae614330e0ae","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Riordan array (1-x,x/(1-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1","6","9"],["2","16","42","36"],["4","40","148","240","144"],["8","96","456","1072","1248","576"],["16","224","1296","3968","6784","6144","2304"],["32","512","3488","13120","29440","39424","29184","9216"],["64","1152","9024","40192","111360","196608","216064","135168","36864"],["128","2560","22656","116480","383488","838656","1218560","1134592","614400","147456"],["256","5632","55552","323584","1232896","3211264","5791744","7143424","5767168","2752512","589824"],["512","12288","133632","869376","3760128","11354112","24428544","37453824","40108032","28573696","12189696","2359296"],["1024","26624","316416","2273280","10997760","37748736","94273536","172621824","230031360","217579520","138674176","53477376","9437184"],["2048","57344","739328","5812224","31088640","119488512","339542016","722337792","1150550016","1355284480","1147666432","661651456","232783872","37748736"],["4096","122880","1708032","14581760","85426176","363331584","1157038080","2802843648","5190451200","7312769024","7716470784","5913968640","3112173568","1006632960","150994944"],["8192","262144","3907584","35995648","229179392","1068367872","3767402496","10233839616","21592276992","35387342848","44684017664","42693820416","29880221696","14461960192","4328521728","603979776"],["16384","557056","8863744","87621632","602341376","3053453312","11808276480","35537289216","84119912448","157143793664","230917406720","264123711488","230535725056","148444807168","66504884224","18522046464","2415919104"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["16","96","576","3456","20736","124416","746496","4478976","26873856","161243136","967458816","5804752896","34828517376","208971104256","1253826625536"],"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","6","9"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178556,"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-0a32d7d9dba0e656"]},{"id":178557,"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-0a32d7d9dba0e656"]},{"id":178558,"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-0a32d7d9dba0e656"]}],"region_exact":{},"incoming_edges":[{"id":"e-0a32d7d9dba0e656","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-26fdd4314504ef0a","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":"473c74a483ee90c9f8f0cf64dce4da4e99a690f092431128505d0dc5d6d427de","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-43cc6f2245a53e97","role":"sequence","name":"basis_map_A013609 of Riordan array (1-x,x/(1-x)).","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-46dfe5ce11de2278","kind":"fingerprint_related","node_a_id":"n-39578f69b35e8349","node_b_id":"n-26fdd4314504ef0a","fingerprint":"f6-063dae614330e0ae","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-063dae614330e0ae"},"other_id":"n-39578f69b35e8349","other":{"id":"n-39578f69b35e8349","name":"basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","first_index":0,"fingerprint":"f6-063dae614330e0ae","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","6","1"],["18","21","8","1"],["36","60","37","10","1"]],"card_rows":[["9","6","1"],["18","21","8","1"],["36","60","37","10","1"],["72","156","134","57","12","1"]],"row_sums_preview":["16","48","144","432","1296","3888","11664","34992"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["x \u21a6 x+1"],"edges":["e-3fc4d53dea6f488e"],"text":"A391533 \u2192 x \u21a6 x+1"},"headline":"A391533 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-651fc116722d5c64","kind":"fingerprint_related","node_a_id":"n-5cb613a96f187602","node_b_id":"n-26fdd4314504ef0a","fingerprint":"f6-063dae614330e0ae","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-063dae614330e0ae"},"other_id":"n-5cb613a96f187602","other":{"id":"n-5cb613a96f187602","name":"mobius_x_over_x_plus_one of reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","first_index":0,"fingerprint":"f6-063dae614330e0ae","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","6","9"],["1","8","21","18"],["1","10","37","60","36"]],"card_rows":[["1","6","9"],["1","8","21","18"],["1","10","37","60","36"],["1","12","57","134","156","72"]],"row_sums_preview":["16","48","144","432","1296","3888","11664","34992"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-6248136e99299279","e-30a6816fcb73f40f"],"text":"A391533 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A391533 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f47768a57cc3633a","kind":"fingerprint_related","node_a_id":"n-4021a45c25536331","node_b_id":"n-26fdd4314504ef0a","fingerprint":"f6-063dae614330e0ae","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-063dae614330e0ae"},"other_id":"n-4021a45c25536331","other":{"id":"n-4021a45c25536331","name":"reversal of basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","first_index":0,"fingerprint":"f6-063dae614330e0ae","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","6","9"],["1","8","21","18"],["1","10","37","60","36"]],"card_rows":[["1","6","9"],["1","8","21","18"],["1","10","37","60","36"],["1","12","57","134","156","72"]],"row_sums_preview":["16","48","144","432","1296","3888","11664","34992"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["x \u21a6 x+1","reverse"],"edges":["e-3fc4d53dea6f488e","e-52d5fa4a5f0809a1"],"text":"A391533 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A391533 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}}],"oeis":[],"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,     6,      9;\n  1:   2,    16,     42,     36;\n  2:   4,    40,    148,    240,     144;\n  3:   8,    96,    456,   1072,    1248,      576;\n  4:  16,   224,   1296,   3968,    6784,     6144,     2304;\n  5:  32,   512,   3488,  13120,   29440,    39424,    29184,     9216;\n  6:  64,  1152,   9024,  40192,  111360,   196608,   216064,   135168,    36864;\n  7: 128,  2560,  22656, 116480,  383488,   838656,  1218560,  1134592,   614400,   147456;\n  8: 256,  5632,  55552, 323584, 1232896,  3211264,  5791744,  7143424,  5767168,  2752512,   589824;\n  9: 512, 12288, 133632, 869376, 3760128, 11354112, 24428544, 37453824, 40108032, 28573696, 12189696, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2+6*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2.","mathematica":"P[0, x_] := 1 + 6*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + 6*x + 4*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 6*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 6*x + 4*x^2)*v[i - 1])/(1 + x)); 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-7717563017cb9e5f","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-4ff089327fda7415","e-0a32d7d9dba0e656"],"text":"A159854 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A159854 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}