{"id":"n-81c9d624be1e751d","revision":0,"first_index":0,"fingerprint":"f6-4b5617cf7459dd0c","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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":[["144","552","529"],["1728","9936","19044","12167"],["20736","158976","457056","584016","279841"],["248832","2384640","9141120","17520480","16790460","6436343"],["2985984","34338816","164540160","420491520","604456560","463416696","148035889"],["35831808","480743424","2764274688","8830321920","16924783680","19463501232","12435014676","3404825447"],["429981696","6593052672","44228395008","169542180864","406194808320","622832039424","596880704448","326863242912","78310985281"],["5159780352","89006211072","682380951552","3051759255552","8773807859712","16816465064448","21487705360128","17650615117248","8457586410348","1801152661463"],["61917364224","1186749480960","10235714273280","52315872952320","175476157194240","403595161546752","644631160803840","706024604689920","507455184620880","216138319375560","41426511213649"],["743008370688","15665093148672","150123809341440","863211903713280","3308978964234240","8879093554028544","17018262645221376","23298811954767360","22328028123318720","14265129078786960","5468299480201668","952809757913927"],["8916100448256","205070310309888","2161782854516736","13811390459412480","59561621356216320","182655638825730048","408438303485313024","671005784297299968","803809012439473920","684726195781774080","393717562574520096","137204605139605488","21914624432020321"],["106993205379072","2665914034028544","30658011391328256","215457691166834688","1032401436841082880","3561784957101735936","9102339334815547392","17446150391729799168","25078841188111586304","26704321635489189120","20473313253875044992","10701959200889228064","3418681411395170076","504036361936467383"],["1283918464548864","34451812132061184","429212159478595584","3290626556002566144","17344344138930192384","66486652532565737472","191149126031126495232","418707609401515180032","702207553267124416512","897265206952436754432","859879156662751889664","599309715249796771584","287169238557194286384","84678108805326520344","11592836324538749809"],["15407021574586368","442951870269358080","5942937592780554240","49359398340038492160","283816540455221329920","1196759745586183274496","3822982520622529904640","9420921211534091550720","18056765655440342138880","26917956208573102632960","30955649639859068027904","26968937186240854721280","17230154313431657183040","7621029792479386830960","2086710538416974965620","266635235464391245607"],["184884258895036416","5669783939447783424","81503144129561886720","729000344714414653440","4541064647283541278720","20888897377504289882112","73401264395952574169088","200979652512727286415360","433362375730568211333120","738321084578005100789760","990580788475490176892928","1035607187951648821297152","827047407044719544785920","487745906718680757181440","200324211688029596699520","51193965209163119156544","6132610415680998648961"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1225","42875","1500625","52521875","1838265625","64339296875","2251875390625","78815638671875","2758547353515625","96549157373046875","3379220508056640625","118272717781982421875","4139545122369384765625","144884079282928466796875","5070942774902496337890625"],"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":[["144","552","529"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["12","35","23"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193779,"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-e963ff8ff2552a07"]},{"id":193780,"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-e963ff8ff2552a07"]},{"id":193781,"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-e963ff8ff2552a07"]}],"region_exact":{},"incoming_edges":[{"id":"e-e963ff8ff2552a07","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-81c9d624be1e751d","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":"ca5373a610a599c89c8976c21a0b3c51021a19261b7f45b94e42de7b37f530bb","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-344e4bfa9933088a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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-2c64686ac9f5ab31","kind":"fingerprint_related","node_a_id":"n-d3d8e71722f9bca4","node_b_id":"n-81c9d624be1e751d","fingerprint":"f6-4b5617cf7459dd0c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b5617cf7459dd0c"},"other_id":"n-d3d8e71722f9bca4","other":{"id":"n-d3d8e71722f9bca4","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-4b5617cf7459dd0c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["12"],["552","288"]],"card_rows":[["12"],["552","288"],["19044","19872","5184"],["584016","914112","476928","82944"]],"row_sums_preview":["0","12","840","44100","2058000","90037500","3781575000","154414312500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["x \u21a6 x+1","D"],"edges":["e-37d2e50d1bfe4331","e-a10c05bf00dc2415"],"text":"A038326 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038326 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-393fdecad648c6e9","kind":"fingerprint_related","node_a_id":"n-81c9d624be1e751d","node_b_id":"n-30a6e10f806704ab","fingerprint":"f6-4b5617cf7459dd0c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b5617cf7459dd0c"},"other_id":"n-30a6e10f806704ab","other":{"id":"n-30a6e10f806704ab","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-4b5617cf7459dd0c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["529","552","144"],["12167","19044","9936","1728"],["279841","584016","457056","158976","20736"]],"card_rows":[["529","552","144"],["12167","19044","9936","1728"],["279841","584016","457056","158976","20736"],["6436343","16790460","17520480","9141120","2384640","248832"]],"row_sums_preview":["1225","42875","1500625","52521875","1838265625","64339296875","2251875390625","78815638671875"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-653ba39db5056185","e-322443d51cb9dce2"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-3ac744d9775b45e6","kind":"fingerprint_related","node_a_id":"n-454026b53ecaf199","node_b_id":"n-81c9d624be1e751d","fingerprint":"f6-4b5617cf7459dd0c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b5617cf7459dd0c"},"other_id":"n-454026b53ecaf199","other":{"id":"n-454026b53ecaf199","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-4b5617cf7459dd0c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","23"],["144","552","529"]],"card_rows":[["1"],["12","23"],["144","552","529"],["1728","9936","19044","12167"]],"row_sums_preview":["1","35","1225","42875","1500625","52521875","1838265625","64339296875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["x \u21a6 x+1","reverse"],"edges":["e-37d2e50d1bfe4331","e-40ec35d33fca402d"],"text":"A038326 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038326 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-960f0547ab1609c2","kind":"fingerprint_related","node_a_id":"n-89f02ea58ff2bc55","node_b_id":"n-81c9d624be1e751d","fingerprint":"f6-4b5617cf7459dd0c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b5617cf7459dd0c"},"other_id":"n-89f02ea58ff2bc55","other":{"id":"n-89f02ea58ff2bc55","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-4b5617cf7459dd0c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["23","12"],["529","552","144"]],"card_rows":[["1"],["23","12"],["529","552","144"],["12167","19044","9936","1728"]],"row_sums_preview":["1","35","1225","42875","1500625","52521875","1838265625","64339296875"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["x \u21a6 x+1"],"edges":["e-37d2e50d1bfe4331"],"text":"A038326 \u2192 x \u21a6 x+1"},"headline":"A038326 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-98a3457f2db5dbfe","kind":"fingerprint_related","node_a_id":"n-81c9d624be1e751d","node_b_id":"n-a1d15ac1f4dec54a","fingerprint":"f6-4b5617cf7459dd0c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b5617cf7459dd0c"},"other_id":"n-a1d15ac1f4dec54a","other":{"id":"n-a1d15ac1f4dec54a","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-4b5617cf7459dd0c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2116","2208","576"],["97336","152352","79488","13824"],["4477456","9344256","7312896","2543616","331776"]],"card_rows":[["2116","2208","576"],["97336","152352","79488","13824"],["4477456","9344256","7312896","2543616","331776"],["205962976","537294720","560655360","292515840","76308480","7962624"]],"row_sums_preview":["4900","343000","24010000","1680700000","117649000000","8235430000000","576480100000000","40353607000000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-61350e73fec19600","e-08dbeb20cf839ce8"],"text":"A038314 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038314 \u2192 basis A013609 \u2192 x \u21a6 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:          144,            552,             529;\n  1:         1728,           9936,           19044,           12167;\n  2:        20736,         158976,          457056,          584016,           279841;\n  3:       248832,        2384640,         9141120,        17520480,         16790460,          6436343;\n  4:      2985984,       34338816,       164540160,       420491520,        604456560,        463416696,         148035889;\n  5:     35831808,      480743424,      2764274688,      8830321920,      16924783680,      19463501232,       12435014676,        3404825447;\n  6:    429981696,     6593052672,     44228395008,    169542180864,     406194808320,     622832039424,      596880704448,      326863242912,       78310985281;\n  7:   5159780352,    89006211072,    682380951552,   3051759255552,    8773807859712,   16816465064448,    21487705360128,    17650615117248,     8457586410348,     1801152661463;\n  8:  61917364224,  1186749480960,  10235714273280,  52315872952320,  175476157194240,  403595161546752,   644631160803840,   706024604689920,   507455184620880,   216138319375560,   41426511213649;\n  9: 743008370688, 15665093148672, 150123809341440, 863211903713280, 3308978964234240, 8879093554028544, 17018262645221376, 23298811954767360, 22328028123318720, 14265129078786960, 5468299480201668, 952809757913927;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (12+35*x+23*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 552*x + 529*x^2.","mathematica":"P[0, x_] := 144 + 552*x + 529*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((12 + 35*x + 23*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] = 144 + 552*x + 529*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 35*x + 23*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-344e4bfa9933088a","steps":["x \u21a6 x/(x+1)"],"edges":["e-e963ff8ff2552a07"],"text":"A038337 \u2192 x \u21a6 x/(x+1)"},"headline":"A038337 \u2192 x \u21a6 x/(x+1)","chain":null}