{"id":"n-656bbcfcd6756619","revision":0,"first_index":0,"fingerprint":"f6-f0770160d0c19011","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","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":[["196","84","9"],["2744","1764","378","27"],["38416","32928","10584","1512","81"],["537824","576240","246960","52920","5670","243"],["7529536","9680832","5186160","1481760","238140","20412","729"],["105413504","158120256","101648736","36303120","7779240","1000188","71442","2187"],["1475789056","2529924096","1897443072","813189888","217818720","37340352","4000752","244944","6561"],["20661046784","39846304512","34153975296","17076987648","5489031744","1176221088","168031584","15431472","826686","19683"],["289254654976","619831403520","597694567680","341539752960","128077407360","32934190464","5881105440","720135360","57868020","2755620","59049"],["4049565169664","9545403614208","10227218158080","6574640244480","2817702961920","845310888576","181138047552","27725211360","2970558360","212182740","9093546","177147"],["56693912375296","145784346107904","171817265055744","122726617896960","59171762200320","20287461325824","5071865331456","931567101696","124763451120","11882233440","763857864","29760696","531441"],["793714773254144","2211062582636544","2842794749104128","2233624445724672","1196584524495360","461539745162496","131868498617856","28257535418112","4541389620768","540641621520","46340710416","2708223336","96722262","1594323"],["11112006825558016","33336020476674048","46432314235367424","39799126487457792","23453056680109056","10051310005761024","3230778216137472","791210991707136","148352060945088","21193151563584","2270694810384","176937257952","9478781676","312487308","4782969"],["155568095557812224","500040307150110720","750060460725166080","696484713530511360","447740172983900160","211077510120981504","75384825043207680","20769288532312320","4450561828352640","741760304725440","95369182036128","9289206042480","663514717320","32811167340","1004423490","14348907"],["2177953337809371136","7467268586774986752","12000967371602657280","12000967371602657280","8357816562366136320","4298305660645441536","1688620080967852032","516924514581995520","124615731193873920","23736329751214080","3560449462682112","416156430703104","37156824169920","2449900494720","112495430880","3214155168","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["289","4913","83521","1419857","24137569","410338673","6975757441","118587876497","2015993900449","34271896307633","582622237229761","9904578032905937","168377826559400929","2862423051509815793","48661191875666868481"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["196","84","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["14","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192003,"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-f127af8a231f24d4"]},{"id":192004,"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-4c12bc1ddcf66c68","e-f127af8a231f24d4"]},{"id":192005,"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-4c12bc1ddcf66c68","e-f127af8a231f24d4"]},{"id":192006,"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-f127af8a231f24d4"]},{"id":192007,"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-4c12bc1ddcf66c68"]}],"region_exact":{},"incoming_edges":[{"id":"e-4c12bc1ddcf66c68","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-656bbcfcd6756619","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":"fdfef90d27db77312fb42c2b066ad44615f14c3a7810bcb7e7b420fd93ef7920","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-809ca598570cd420","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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-f127af8a231f24d4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-656bbcfcd6756619","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":"50639752b60473a46ae5684efddb87b86e19a43a9abdf9ab3da28caf4c6bcd45","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-809ca598570cd420","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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-740b6b978808895c","kind":"fingerprint_related","node_a_id":"n-087fd98ce32430ab","node_b_id":"n-656bbcfcd6756619","fingerprint":"f6-f0770160d0c19011","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f0770160d0c19011"},"other_id":"n-087fd98ce32430ab","other":{"id":"n-087fd98ce32430ab","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-f0770160d0c19011","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","3"],["196","84","9"]],"card_rows":[["1"],["14","3"],["196","84","9"],["2744","1764","378","27"]],"row_sums_preview":["1","17","289","4913","83521","1419857","24137569","410338673"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610","basis A007318"],"edges":["e-254f1b828fe6620e","e-278bc679e566435a"],"text":"A038303 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038303 \u2192 basis A013610 \u2192 basis A007318","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:           196,            84,              9;\n  1:          2744,          1764,            378,            27;\n  2:         38416,         32928,          10584,          1512,            81;\n  3:        537824,        576240,         246960,         52920,          5670,          243;\n  4:       7529536,       9680832,        5186160,       1481760,        238140,        20412,          729;\n  5:     105413504,     158120256,      101648736,      36303120,       7779240,      1000188,        71442,        2187;\n  6:    1475789056,    2529924096,     1897443072,     813189888,     217818720,     37340352,      4000752,      244944,       6561;\n  7:   20661046784,   39846304512,    34153975296,   17076987648,    5489031744,   1176221088,    168031584,    15431472,     826686,     19683;\n  8:  289254654976,  619831403520,   597694567680,  341539752960,  128077407360,  32934190464,   5881105440,   720135360,   57868020,   2755620,   59049;\n  9: 4049565169664, 9545403614208, 10227218158080, 6574640244480, 2817702961920, 845310888576, 181138047552, 27725211360, 2970558360, 212182740, 9093546, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 84*x + 9*x^2.","mathematica":"P[0, x_] := 196 + 84*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(14 + 3*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] = 196 + 84*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 3*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-108b34f10b7468f4","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-3a9169eda66e0655","e-4c12bc1ddcf66c68"],"text":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}
