{"id":"n-e8edca1dfa93add7","revision":0,"first_index":0,"fingerprint":"f6-0e62401a26557dd3","name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^n.","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"],["13","8"],["169","208","64"],["2197","4056","2496","512"],["28561","70304","64896","26624","4096"],["371293","1142440","1406080","865280","266240","32768"],["4826809","17822064","27418560","22497280","10383360","2555904","262144"],["62748517","270301304","499017792","511813120","314961920","116293632","23855104","2097152"],["815730721","4015905088","8649641728","10645712896","8189009920","4031512576","1240465408","218103808","16777216"],["10604499373","58732611912","144572583168","207591401472","191622832128","117921742848","48378150912","12759072768","1962934272","134217728"],["137858491849","848359949840","2349304476480","3855268884480","4151828029440","3065965314048","1572289904640","552893153280","127590727680","17448304640","1073741824"],["1792160394037","12131547282712","37327837792960","68912931310080","84815915458560","73072173318144","44967491272704","19765930229760","6081824686080","1247553781760","153545080832","8589934592"],["23298085122481","172047397827552","582314269570176","1194490809374720","1653910351441920","1628465576804352","1169154773090304","616697023168512","237191162757120","64872796651520","11976516304896","1340029796352","68719476736"],["302875106592253","2423000852738024","8946464687032704","20186894678432768","31056761043742720","34401335309991936","28226736664608768","17370299485913088","8017061301190656","2740875658526720","674677085175808","113232517791744","11613591568384","549755813888"],["3937376385699289","33922011938332336","135688047753329344","334001348315887616","565233050996117504","695671447379836928","642158259119849472","451627786633740288","243184192802783232","99767873970372608","30697807375499264","6869439412699136","1056836832722944","100055558127616","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","21","441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581","16679880978201","350277500542221","7355827511386641","154472377739119461","3243919932521508681"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["13","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186978,"position":0,"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-45e799a50986c859"]},{"id":186979,"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-0383134140c02266","e-45e799a50986c859","e-eab328c98801dd53","e-f4fa0a8b4b5be496"]},{"id":186980,"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-0383134140c02266","e-45e799a50986c859","e-eab328c98801dd53","e-f4fa0a8b4b5be496"]},{"id":186981,"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-45e799a50986c859"]},{"id":186982,"position":4,"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-0383134140c02266","e-eab328c98801dd53","e-f4fa0a8b4b5be496"]}],"region_exact":{},"incoming_edges":[{"id":"e-0383134140c02266","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-e8edca1dfa93add7","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":"d3dd23eae0a4013b81a0067a41830a6b4365aa4f7f62ba6fa30133c01dcf65ec","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-210f23d82ddfabd8","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^n.","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-eab328c98801dd53","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-e8edca1dfa93add7","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":"3e35e24167eff66ed3c16addcbf3250dc113afee551df4d4695edd6f297d0141","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-10f70e3ea8b205cf","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^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]"}]},{"id":"e-f4fa0a8b4b5be496","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-e8edca1dfa93add7","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":"7738a7b262c3ac78ee75edf35da2803ce1daf288b570fc24662f0d658698a35a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b8c0cc94a00a4dd8","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^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]"}]},{"id":"e-45e799a50986c859","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-e8edca1dfa93add7","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":"c07e54956a9d1f9e0927512c06e7d2c020c6fde9c78e1c141413b707ccb803f0","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-210f23d82ddfabd8","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^n.","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-450dae05e955f761","kind":"fingerprint_related","node_a_id":"n-e8edca1dfa93add7","node_b_id":"n-c443a0508165a48c","fingerprint":"f6-0e62401a26557dd3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0e62401a26557dd3"},"other_id":"n-c443a0508165a48c","other":{"id":"n-c443a0508165a48c","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-0e62401a26557dd3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","832","676"],["4096","19968","32448","17576"],["65536","425984","1038336","1124864","456976"]],"card_rows":[["256","832","676"],["4096","19968","32448","17576"],["65536","425984","1038336","1124864","456976"],["1048576","8519680","27688960","44994560","36558080","11881376"]],"row_sums_preview":["1764","74088","3111696","130691232","5489031744","230539333248","9682651996416","406671383849472"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-772e88d8a50e6734","e-bb602c72254744fe"],"text":"A038319 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038319 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-46d9dcc3e4c89f10","kind":"fingerprint_related","node_a_id":"n-e8edca1dfa93add7","node_b_id":"n-60e14c585778a1ea","fingerprint":"f6-0e62401a26557dd3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0e62401a26557dd3"},"other_id":"n-60e14c585778a1ea","other":{"id":"n-60e14c585778a1ea","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-0e62401a26557dd3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","13"],["64","208","169"]],"card_rows":[["1"],["8","13"],["64","208","169"],["512","2496","4056","2197"]],"row_sums_preview":["1","21","441","9261","194481","4084101","85766121","1801088541"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-31008ec5d02c6da5","e-461c166da3c28c86"],"text":"A013622 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A013622 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","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:          13,           8;\n  2:         169,         208,           64;\n  3:        2197,        4056,         2496,          512;\n  4:       28561,       70304,        64896,        26624,         4096;\n  5:      371293,     1142440,      1406080,       865280,       266240,        32768;\n  6:     4826809,    17822064,     27418560,     22497280,     10383360,      2555904,      262144;\n  7:    62748517,   270301304,    499017792,    511813120,    314961920,    116293632,    23855104,     2097152;\n  8:   815730721,  4015905088,   8649641728,  10645712896,   8189009920,   4031512576,  1240465408,   218103808,   16777216;\n  9: 10604499373, 58732611912, 144572583168, 207591401472, 191622832128, 117921742848, 48378150912, 12759072768, 1962934272, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (13+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(13 + 8*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; for(n = 1, N, my(i = n + 1); v[i] = (13 + 8*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-01cea33b6485d9cc","steps":["basis A013611","basis A007318"],"edges":["e-391376c704e767eb","e-0383134140c02266"],"text":"A038220 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A038220 \u2192 basis A013611 \u2192 basis A007318","chain":null}