{"id":"n-61e603a5421ea593","revision":0,"first_index":0,"fingerprint":"f6-e348a50f4e54dede","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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":true,"rows":[["1"],["8","11"],["64","176","121"],["512","2112","2904","1331"],["4096","22528","46464","42592","14641"],["32768","225280","619520","851840","585640","161051"],["262144","2162688","7434240","13629440","14055360","7730448","1771561"],["2097152","20185088","83263488","190812160","262366720","216452544","99207416","19487171"],["16777216","184549376","888143872","2442395648","4197867520","4617654272","3174637312","1247178944","214358881"],["134217728","1660944384","9135194112","29308747776","60449292288","83117776896","76191295488","44898441984","15433839432","2357947691"],["1073741824","14763950080","91351941120","334957117440","805990563840","1329884430336","1523825909760","1197291786240","617353577280","188635815280","25937424601"],["8589934592","129922760704","893218979840","3684528291840","10132452802560","19504971644928","26819336011776","26340419297280","18109038266880","8299975872320","2282493364888","285311670611"],["68719476736","1133871366144","8574902206464","39301635112960","121589433630720","267496753987584","429109376188416","505736050507776","434616918405120","265599227914240","109559681514624","27389920378656","3138428376721"],["549755813888","9826885173248","81071802679296","408737005174784","1405033455288320","3477457801838592","6375339303370752","8766091542134784","9040031902826496","6905579925770240","3798068959173632","1424275859690112","326396551178984","34522712143931"],["4398046511104","84662395338752","756670158340096","4161685870870528","15736374699229184","43275030422880256","89254750247190528","140257464674156544","168747262186094592","154684990337253376","106345930856861696","53172965428430848","18278206866023104","3866543760120272","379749833583241"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","19","361","6859","130321","2476099","47045881","893871739","16983563041","322687697779","6131066257801","116490258898219","2213314919066161","42052983462257059","799006685782884121"],"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":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["8","19","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186031,"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-e5ada0297e914fac"]},{"id":186032,"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-e5ada0297e914fac"]},{"id":186033,"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-e5ada0297e914fac"]}],"region_exact":{},"incoming_edges":[{"id":"e-e5ada0297e914fac","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-61e603a5421ea593","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":"f329e0121fb9173462b88734256296d4becf130b902354d74fae0ea2d3a00b89","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-a227586ec36efe06","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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-101a109ff338e070","kind":"candidate_equal","node_a_id":"n-61e603a5421ea593","node_b_id":"n-8bf6259feffcb309","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-8bf6259feffcb309","other":{"id":"n-8bf6259feffcb309","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","11"],["64","176","121"]],"card_rows":[["1"],["8","11"],["64","176","121"],["512","2112","2904","1331"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["reverse"],"edges":["e-5a9754a240da486b"],"text":"A038322 \u2192 reverse"},"headline":"A038322 \u2192 reverse","chain":null}},{"id":"l-270067acd0222481","kind":"fingerprint_related","node_a_id":"n-2f02b2035f8f4541","node_b_id":"n-61e603a5421ea593","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-2f02b2035f8f4541","other":{"id":"n-2f02b2035f8f4541","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","8"],["121","176","64"]],"card_rows":[["1"],["11","8"],["121","176","64"],["1331","2904","2112","512"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038322"],"derivation":{"root":"n-2f02b2035f8f4541","steps":[],"edges":[],"text":"A038322"},"headline":"A038322: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","chain":null}},{"id":"l-9b2ea87fcdabcc8f","kind":"fingerprint_related","node_a_id":"n-61e603a5421ea593","node_b_id":"n-8381edfecc6c9677","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-8381edfecc6c9677","other":{"id":"n-8381edfecc6c9677","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4096","11264","7744"],["262144","1081344","1486848","681472"],["16777216","92274688","190316544","174456832","59969536"]],"card_rows":[["4096","11264","7744"],["262144","1081344","1486848","681472"],["16777216","92274688","190316544","174456832","59969536"],["1073741824","7381975040","20300431360","27913093120","19190251520","5277319168"]],"row_sums_preview":["23104","3511808","533794816","81136812032","12332795428864","1874584905187328","284936905588473856","43310409649448026112"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["basis A013611","basis A013609"],"edges":["e-9ac6ce8d07a161c9","e-5898adfc55563eae"],"text":"A038301 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038301 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-a2e2120b2c925573","kind":"fingerprint_related","node_a_id":"n-2e0a8214849adcd6","node_b_id":"n-61e603a5421ea593","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-2e0a8214849adcd6","other":{"id":"n-2e0a8214849adcd6","name":"basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","176","64"],["1331","2904","2112","512"],["14641","42592","46464","22528","4096"]],"card_rows":[["121","176","64"],["1331","2904","2112","512"],["14641","42592","46464","22528","4096"],["161051","585640","851840","619520","225280","32768"]],"row_sums_preview":["361","6859","130321","2476099","47045881","893871739","16983563041","322687697779"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A007318"],"edges":["e-0566c972c7be0ca0"],"text":"A038226 \u2192 basis A007318"},"headline":"A038226 \u2192 basis A007318","chain":null}},{"id":"l-a8654c5eab25cb6d","kind":"fingerprint_related","node_a_id":"n-61e603a5421ea593","node_b_id":"n-d8b9bf331a6f183a","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-d8b9bf331a6f183a","other":{"id":"n-d8b9bf331a6f183a","name":"derivative of basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["176","128"],["2904","4224","1536"],["42592","92928","67584","16384"]],"card_rows":[["176","128"],["2904","4224","1536"],["42592","92928","67584","16384"],["585640","1703680","1858560","901120","163840"]],"row_sums_preview":["304","8664","219488","5212840","118852752","2634569336","57207791296","1222816538952"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A007318","D"],"edges":["e-0566c972c7be0ca0","e-0ec9ff8fdacbcaa0"],"text":"A038226 \u2192 basis A007318 \u2192 D"},"headline":"A038226 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-bfca102541f617e5","kind":"fingerprint_related","node_a_id":"n-61e603a5421ea593","node_b_id":"n-b6d91d50fb3f7b0a","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-b6d91d50fb3f7b0a","other":{"id":"n-b6d91d50fb3f7b0a","name":"reversal of basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["64","176","121"],["512","2112","2904","1331"],["4096","22528","46464","42592","14641"]],"card_rows":[["64","176","121"],["512","2112","2904","1331"],["4096","22528","46464","42592","14641"],["32768","225280","619520","851840","585640","161051"]],"row_sums_preview":["361","6859","130321","2476099","47045881","893871739","16983563041","322687697779"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A007318","reverse"],"edges":["e-0566c972c7be0ca0","e-305b6a6c40612ef6"],"text":"A038226 \u2192 basis A007318 \u2192 reverse"},"headline":"A038226 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-d4556778db86a27a","kind":"fingerprint_related","node_a_id":"n-61e603a5421ea593","node_b_id":"n-b3d9b43af875a5c4","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-b3d9b43af875a5c4","other":{"id":"n-b3d9b43af875a5c4","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["64","176","121"],["512","2112","2904","1331"],["4096","22528","46464","42592","14641"]],"card_rows":[["64","176","121"],["512","2112","2904","1331"],["4096","22528","46464","42592","14641"],["32768","225280","619520","851840","585640","161051"]],"row_sums_preview":["361","6859","130321","2476099","47045881","893871739","16983563041","322687697779"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["x \u21a6 x/(x+1)"],"edges":["e-175134310e4c9ec8"],"text":"A038281 \u2192 x \u21a6 x/(x+1)"},"headline":"A038281 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-fccd0fe15bc9cfeb","kind":"fingerprint_related","node_a_id":"n-61e603a5421ea593","node_b_id":"n-6f78dd889d006015","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-6f78dd889d006015","other":{"id":"n-6f78dd889d006015","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["176","128"]],"card_rows":[["8"],["176","128"],["2904","4224","1536"],["42592","92928","67584","16384"]],"row_sums_preview":["0","8","304","8664","219488","5212840","118852752","2634569336"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["D"],"edges":["e-6829052708e58dc8"],"text":"A038322 \u2192 D"},"headline":"A038322 \u2192 D","chain":null}},{"id":"l-fd5d1e176e1880cf","kind":"fingerprint_related","node_a_id":"n-61e603a5421ea593","node_b_id":"n-e6ce82432a2d62f1","fingerprint":"f6-e348a50f4e54dede","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e348a50f4e54dede"},"other_id":"n-e6ce82432a2d62f1","other":{"id":"n-e6ce82432a2d62f1","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-e348a50f4e54dede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","704","256"],["10648","23232","16896","4096"],["234256","681472","743424","360448","65536"]],"card_rows":[["484","704","256"],["10648","23232","16896","4096"],["234256","681472","743424","360448","65536"],["5153632","18740480","27258880","19824640","7208960","1048576"]],"row_sums_preview":["1444","54872","2085136","79235168","3010936384","114415582592","4347792138496","165216101262848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013611","basis A013609"],"edges":["e-0eb3975e7d8d74fc","e-08b8ba2164b96d10"],"text":"A038328 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038328 \u2192 basis A013611 \u2192 basis A013609","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;\n  1:         8,         11;\n  2:        64,        176,        121;\n  3:       512,       2112,       2904,        1331;\n  4:      4096,      22528,      46464,       42592,       14641;\n  5:     32768,     225280,     619520,      851840,      585640,      161051;\n  6:    262144,    2162688,    7434240,    13629440,    14055360,     7730448,     1771561;\n  7:   2097152,   20185088,   83263488,   190812160,   262366720,   216452544,    99207416,    19487171;\n  8:  16777216,  184549376,  888143872,  2442395648,  4197867520,  4617654272,  3174637312,  1247178944,   214358881;\n  9: 134217728, 1660944384, 9135194112, 29308747776, 60449292288, 83117776896, 76191295488, 44898441984, 15433839432, 2357947691;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (8+19*x+11*x^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[((8 + 19*x + 11*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; for(n = 1, N, my(i = n + 1); v[i] = ((8 + 19*x + 11*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-f211cb6153c37c6d","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-892638ce5a781886","e-e5ada0297e914fac"],"text":"A027466 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A027466 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}