{"id":"n-c4b1d536a6354c0e","revision":0,"first_index":0,"fingerprint":"f6-dde9c6f2b382eac4","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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":[["36","204","289"],["216","1836","5202","4913"],["1296","14688","62424","117912","83521"],["7776","110160","624240","1768680","2505630","1419857"],["46656","793152","5618160","21224160","45101340","51114852","24137569"],["279936","5552064","47192544","222853680","631418760","1073411892","1013777898","410338673"],["1679616","38071296","377540352","2139395328","7577025120","17174590272","24330669552","19696256304","6975757441"],["10077696","256981248","2912454144","19254557952","81831871296","231856968672","437952051936","531798920208","376690901814","118587876497"],["60466176","1713208320","21843406080","165039068160","818318712960","2782283624064","6569280779040","10635978404160","11300727054420","7115272589820","2015993900449"],["362797056","11307174912","160184977920","1361572312320","7715576436480","30605119864704","86714506283328","175493643668640","248615995197240","234803995464060","133055597429634","34271896307633"],["2176782336","74010599424","1153331841024","10892578498560","69440187928320","314795518608384","1040574075399936","2527108468828416","4475087913550320","5635295891137440","4790001507466824","2467576534149576","582622237229761"],["13060694016","481068896256","8178171236352","84962112288768","601814962045440","3069256306431744","11594968268742144","32852410094769408","69811371451384992","109888269877180080","124540039194137424","96235484831833464","45444534503921358","9904578032905937"],["78364164096","3108445175808","57247198654464","648801584750592","5055245681181696","28646392193362944","121747166821792512","394228921137232896","977359200319389888","1846122933936625344","2615340823076885904","2694593575291336992","1908670449164697036","831984554764098708","168377826559400929"],["470184984576","19982861844480","396326759915520","4866011885629440","41361101027850240","257817529740266496","1217471668217925120","4435075362793870080","12566046861249298560","27691844009049380160","47076134815383946272","60628355444055082320","57260113474940911080","37439304964384441860","15154004390346083610","2862423051509815793"],["2821109907456","127890315804672","2717669210849280","35933626232340480","330888808222801920","2250043895915053056","11687728014892081152","47307470536467947520","150792562334991582720","379773860695534356480","753218157046143140352","1164064424525857580544","1374242723398581865920","1198057758860302139520","727392210736612013280","274792612944942316128","48661191875666868481"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463","41426511213649","952809757913927","21914624432020321","504036361936467383","11592836324538749809","266635235464391245607","6132610415680998648961"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["36","204","289"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","17"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193848,"position":0,"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-9f8669c49473f56c"]},{"id":193849,"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-9f8669c49473f56c"]},{"id":193850,"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-9f8669c49473f56c"]}],"region_exact":{},"incoming_edges":[{"id":"e-9f8669c49473f56c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-c4b1d536a6354c0e","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"ea3ebd87e24e89a93115954252f1cbf171ccb3c1908bb6a8b00f4f06327f8fd9","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-6ffee7a877aa82b8","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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-0a3ab41739f43d92","kind":"fingerprint_related","node_a_id":"n-c4b1d536a6354c0e","node_b_id":"n-cc7b9cf6ff870c45","fingerprint":"f6-dde9c6f2b382eac4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dde9c6f2b382eac4"},"other_id":"n-cc7b9cf6ff870c45","other":{"id":"n-cc7b9cf6ff870c45","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-dde9c6f2b382eac4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","816","1156"],["1728","14688","41616","39304"],["20736","235008","998784","1886592","1336336"]],"card_rows":[["144","816","1156"],["1728","14688","41616","39304"],["20736","235008","998784","1886592","1336336"],["248832","3525120","19975680","56597760","80180160","45435424"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x/(x+1)","x \u21a6 x/(x+1)"],"edges":["e-492aa71fabef2dc6","e-57727b69da35cbc3"],"text":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 x \u21a6 x/(x+1)"},"headline":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-296a9a82ec403600","kind":"fingerprint_related","node_a_id":"n-c4b1d536a6354c0e","node_b_id":"n-fe432be4158ab93d","fingerprint":"f6-dde9c6f2b382eac4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dde9c6f2b382eac4"},"other_id":"n-fe432be4158ab93d","other":{"id":"n-fe432be4158ab93d","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-dde9c6f2b382eac4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1156","816","144"],["39304","41616","14688","1728"],["1336336","1886592","998784","235008","20736"]],"card_rows":[["1156","816","144"],["39304","41616","14688","1728"],["1336336","1886592","998784","235008","20736"],["45435424","80180160","56597760","19975680","3525120","248832"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-e77df5b989acd52a","e-05bd54587e660aeb"],"text":"A038314 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A038314 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}},{"id":"l-8d898ede591ed954","kind":"fingerprint_related","node_a_id":"n-6ffee7a877aa82b8","node_b_id":"n-c4b1d536a6354c0e","fingerprint":"f6-dde9c6f2b382eac4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dde9c6f2b382eac4"},"other_id":"n-6ffee7a877aa82b8","other":{"id":"n-6ffee7a877aa82b8","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-dde9c6f2b382eac4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["289","204","36"],["4913","5202","1836","216"],["83521","117912","62424","14688","1296"]],"card_rows":[["289","204","36"],["4913","5202","1836","216"],["83521","117912","62424","14688","1296"],["1419857","2505630","1768680","624240","110160","7776"]],"row_sums_preview":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":["x \u21a6 x+1"],"edges":["e-653ba39db5056185"],"text":"A038320 \u2192 x \u21a6 x+1"},"headline":"A038320 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-b6df30d935de816b","kind":"fingerprint_related","node_a_id":"n-c4b1d536a6354c0e","node_b_id":"n-e3503f969f5755f9","fingerprint":"f6-dde9c6f2b382eac4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dde9c6f2b382eac4"},"other_id":"n-e3503f969f5755f9","other":{"id":"n-e3503f969f5755f9","name":"basis_map_A013610 of basis_map_A013609 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-dde9c6f2b382eac4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["17","6"],["289","204","36"]],"card_rows":[["1"],["17","6"],["289","204","36"],["4913","5202","1836","216"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013609","basis A013610"],"edges":["e-267357ec4c02d3cf","e-a4f03ae8d0e08c58"],"text":"A147716 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A147716 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-dd03cd4c55d55034","kind":"fingerprint_related","node_a_id":"n-ef5644e239cb13e5","node_b_id":"n-c4b1d536a6354c0e","fingerprint":"f6-dde9c6f2b382eac4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dde9c6f2b382eac4"},"other_id":"n-ef5644e239cb13e5","other":{"id":"n-ef5644e239cb13e5","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-dde9c6f2b382eac4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["204","72"],["5202","3672","648"],["117912","124848","44064","5184"]],"card_rows":[["204","72"],["5202","3672","648"],["117912","124848","44064","5184"],["2505630","3537360","1872720","440640","38880"]],"row_sums_preview":["276","9522","292008","8395230","231708348","6217507338","163431621456","4228793205174"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":["x \u21a6 x+1","D"],"edges":["e-653ba39db5056185","e-f6ea22cc936e59ea"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 D","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:        36,         204,          289;\n  1:       216,        1836,         5202,          4913;\n  2:      1296,       14688,        62424,        117912,         83521;\n  3:      7776,      110160,       624240,       1768680,       2505630,        1419857;\n  4:     46656,      793152,      5618160,      21224160,      45101340,       51114852,       24137569;\n  5:    279936,     5552064,     47192544,     222853680,     631418760,     1073411892,     1013777898,       410338673;\n  6:   1679616,    38071296,    377540352,    2139395328,    7577025120,    17174590272,    24330669552,     19696256304,      6975757441;\n  7:  10077696,   256981248,   2912454144,   19254557952,   81831871296,   231856968672,   437952051936,    531798920208,    376690901814,    118587876497;\n  8:  60466176,  1713208320,  21843406080,  165039068160,  818318712960,  2782283624064,  6569280779040,  10635978404160,  11300727054420,   7115272589820,   2015993900449;\n  9: 362797056, 11307174912, 160184977920, 1361572312320, 7715576436480, 30605119864704, 86714506283328, 175493643668640, 248615995197240, 234803995464060, 133055597429634, 34271896307633;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 204*x + 289*x^2.","mathematica":"P[0, x_] := 36 + 204*x + 289*x^2;\nP[n_, x_] := P[n, x] = Expand[(6 + 17*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] = 36 + 204*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 17*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-1f2dec04776afb2b","steps":["x \u21a6 x+1","reverse"],"edges":["e-653ba39db5056185","e-9f8669c49473f56c"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}