{"id":"n-2d67e83eace19da6","revision":0,"first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","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)*7^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":[["324","1152","1024"],["5832","31104","55296","32768"],["104976","746496","1990656","2359296","1048576"],["1889568","16796160","59719680","106168320","94371840","33554432"],["34012224","362797056","1612431360","3822059520","5096079360","3623878656","1073741824"],["612220032","7618738176","40633270272","120394874880","214035333120","228304355328","135291469824","34359738368"],["11019960576","156728328192","975198486528","3467372396544","7705271992320","10958609055744","9740985827328","4947802324992","1099511627776"],["198359290368","3173748645888","22568879259648","93619054706688","249650812551168","443823666757632","526013234675712","400771988324352","178120883699712","35184372088832"],["3570467226624","63474972917760","507799783342080","2407347121029120","7489524376535040","15977652003274752","23670595560407040","24046319299461120","16030879532974080","6333186975989760","1125899906842624"],["64268410079232","1256804463771648","11171595233525760","59581841245470720","211846546650562560","527262516108066816","937355584192118784","1190292805323325440","1058038049176289280","626985510622986240","222928181554839552","36028797018963968"],["1156831381426176","24679069470425088","241306457044156416","1429964189891297280","5719856759565189120","16269814782763204608","33744801030916276224","51420649189967659008","57134054655519621120","45142956764855009280","24076243607922671616","7782220156096217088","1152921504606846976"],["20822964865671168","481241854673289216","5133246449848418304","33461162043456356352","148716275748694917120","475892082395823734784","1128040491604915519488","2005405318408738701312","2673873757878318268416","2640862970744018042880","1877947001417968386048","910519758263257399296","269783632078002192384","36893488147419103232"],["374813367582081024","9328688259820683264","107798175446816784384","766564803177363800064","3747650148867111911424","13324978307083064573952","35533275485554838863872","72194591462714593247232","112302697830889367273472","133099493725498509361152","118310661089332008321024","76483659694113621540864","33992737641828276240384","9297159013149614014464","1180591620717411303424"],["6746640616477458432","179910416439398891520","2238885182356963983360","17247708071490685501440","91987776381283656007680","359774414291242743496704","1065998264566645165916160","2436567461866617522094080","4331675487762875594833920","5989477217647432921251840","6388775698823928449335296","5162647029352669454008320","3059346387764544861634560","1255116466775197891952640","318759737593701051924480","37778931862957161709568"],["121439531096594251776","3454279995636458717184","46057066608486116229120","382103071122255186493440","2207706633150807744184320","9419548301443446375186432","30700750019519380778385408","77970158779731760707010560","155940317559463521414021120","246424205526065811617218560","306661233543548565568094208","297368468890713760550879232","220272939919047230037688320","120491180810418997627453440","45901402213492951477125120","10880332376531662572355584","1208925819614629174706176"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2500","125000","6250000","312500000","15625000000","781250000000","39062500000000","1953125000000000","97656250000000000","4882812500000000000","244140625000000000000","12207031250000000000000","610351562500000000000000","30517578125000000000000000","1525878906250000000000000000"],"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":[["324","1152","1024"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["18","50","32"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194379,"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-1095a3d33bcc57bd"]},{"id":194380,"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-1095a3d33bcc57bd"]},{"id":194381,"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-1095a3d33bcc57bd"]}],"region_exact":{},"incoming_edges":[{"id":"e-1095a3d33bcc57bd","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-2d67e83eace19da6","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":"885944b275c3d188e75146aa75174dd5250ad1f1b7e0921615bfe9f466984f62","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-53dc6d1eeb1907f4","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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-091e1093c43af59a","kind":"fingerprint_related","node_a_id":"n-cdff05d2922e65c0","node_b_id":"n-2d67e83eace19da6","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-cdff05d2922e65c0","other":{"id":"n-cdff05d2922e65c0","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["48","27"],["2304","2592","729"]],"card_rows":[["1"],["48","27"],["2304","2592","729"],["110592","186624","104976","19683"]],"row_sums_preview":["1","75","5625","421875","31640625","2373046875","177978515625","13348388671875"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013610","basis A007318"],"edges":["e-b7b3e04e23eb02df","e-3eb2f851fa4ef1f4"],"text":"A038335 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038335 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-47b722a07c692993","kind":"fingerprint_related","node_a_id":"n-2d67e83eace19da6","node_b_id":"n-d5e838559a8c8a11","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-d5e838559a8c8a11","other":{"id":"n-d5e838559a8c8a11","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1024","1152","324"],["32768","55296","31104","5832"],["1048576","2359296","1990656","746496","104976"]],"card_rows":[["1024","1152","324"],["32768","55296","31104","5832"],["1048576","2359296","1990656","746496","104976"],["33554432","94371840","106168320","59719680","16796160","1889568"]],"row_sums_preview":["2500","125000","6250000","312500000","15625000000","781250000000","39062500000000","1953125000000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013610","basis A007318"],"edges":["e-89b2e37c644537dd","e-0f6f6478a7523569"],"text":"A038284 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038284 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-68e0f033d133cbec","kind":"fingerprint_related","node_a_id":"n-2d67e83eace19da6","node_b_id":"n-6e6aac684bd07a75","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-6e6aac684bd07a75","other":{"id":"n-6e6aac684bd07a75","name":"basis_map_A013611 of basis_map_A007318 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","16"],["81","288","256"]],"card_rows":[["1"],["9","16"],["81","288","256"],["729","3888","6912","4096"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["basis A007318","basis A013611"],"edges":["e-13c78408b476aecd","e-9b7b26e5d4ce6a74"],"text":"A013611 \u2192 basis A007318 \u2192 basis A013611"},"headline":"A013611 \u2192 basis A007318 \u2192 basis A013611","chain":null}},{"id":"l-d7f7887586be4509","kind":"fingerprint_related","node_a_id":"n-2d67e83eace19da6","node_b_id":"n-44e3ef545532be70","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-44e3ef545532be70","other":{"id":"n-44e3ef545532be70","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","288","81"],["4096","6912","3888","729"],["65536","147456","124416","46656","6561"]],"card_rows":[["256","288","81"],["4096","6912","3888","729"],["65536","147456","124416","46656","6561"],["1048576","2949120","3317760","1866240","524880","59049"]],"row_sums_preview":["625","15625","390625","9765625","244140625","6103515625","152587890625","3814697265625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-6056f3721ff524be","e-4676c68faaec715a"],"text":"A038305 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038305 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}},{"id":"l-e21d35ad6a2fcff7","kind":"fingerprint_related","node_a_id":"n-2d67e83eace19da6","node_b_id":"n-b2dfc4991f197cd3","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-b2dfc4991f197cd3","other":{"id":"n-b2dfc4991f197cd3","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","9"],["256","288","81"]],"card_rows":[["1"],["16","9"],["256","288","81"],["4096","6912","3888","729"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013610","basis A013610"],"edges":["e-00e3cfe53baaea9b","e-ce295775dae4b69c"],"text":"A038327 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038327 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-e51132f30e741552","kind":"fingerprint_related","node_a_id":"n-2d67e83eace19da6","node_b_id":"n-824192ee04ae06b8","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-824192ee04ae06b8","other":{"id":"n-824192ee04ae06b8","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["729","2592","2304"],["19683","104976","186624","110592"],["531441","3779136","10077696","11943936","5308416"]],"card_rows":[["729","2592","2304"],["19683","104976","186624","110592"],["531441","3779136","10077696","11943936","5308416"],["14348907","127545840","453496320","806215680","716636160","254803968"]],"row_sums_preview":["5625","421875","31640625","2373046875","177978515625","13348388671875","1001129150390625","75084686279296875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A013609","basis A013610"],"edges":["e-c8a2ee9cf24a53ed","e-487092da83f5a6cb"],"text":"A038226 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038226 \u2192 basis A013609 \u2192 basis A013610","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:            324,             1152,              1024;\n  1:           5832,            31104,             55296,             32768;\n  2:         104976,           746496,           1990656,           2359296,            1048576;\n  3:        1889568,         16796160,          59719680,         106168320,           94371840,           33554432;\n  4:       34012224,        362797056,        1612431360,        3822059520,         5096079360,         3623878656,         1073741824;\n  5:      612220032,       7618738176,       40633270272,      120394874880,       214035333120,       228304355328,       135291469824,         34359738368;\n  6:    11019960576,     156728328192,      975198486528,     3467372396544,      7705271992320,     10958609055744,      9740985827328,       4947802324992,       1099511627776;\n  7:   198359290368,    3173748645888,    22568879259648,    93619054706688,    249650812551168,    443823666757632,    526013234675712,     400771988324352,     178120883699712,     35184372088832;\n  8:  3570467226624,   63474972917760,   507799783342080,  2407347121029120,   7489524376535040,  15977652003274752,  23670595560407040,   24046319299461120,   16030879532974080,   6333186975989760,   1125899906842624;\n  9: 64268410079232, 1256804463771648, 11171595233525760, 59581841245470720, 211846546650562560, 527262516108066816, 937355584192118784, 1190292805323325440, 1058038049176289280, 626985510622986240, 222928181554839552, 36028797018963968;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (18+50*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1152*x + 1024*x^2.","mathematica":"P[0, x_] := 324 + 1152*x + 1024*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((18 + 50*x + 32*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] = 324 + 1152*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + 50*x + 32*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-7dcf11eddf2b9a8a","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-322a2e4c0bbd3231","e-1095a3d33bcc57bd"],"text":"A038321 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038321 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}