{"id":"n-3ccd0708169ccfae","revision":1,"first_index":0,"fingerprint":"f6-b1029f17d1b64336","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(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":[["225","270","81"],["3375","6075","3645","729"],["50625","121500","109350","43740","6561"],["759375","2278125","2733750","1640250","492075","59049"],["11390625","41006250","61509375","49207500","22143375","5314410","531441"],["170859375","717609375","1291696875","1291696875","775018125","279006525","55801305","4782969"],["2562890625","12301875000","25833937500","31000725000","23250543750","11160261000","3348078300","573956280","43046721"],["38443359375","207594140625","498225937500","697516312500","627764681250","376658808750","150663523500","38742048900","5811307335","387420489"],["576650390625","3459902343750","9341736328125","14946778125000","15694117031250","11299764262500","5649882131250","1937102445000","435848050125","58113073350","3486784401"],["8649755859375","57088388671875","171265166015625","308277298828125","369932758593750","310743517218750","186446110331250","79905475856250","23971642756875","4794328551375","575319426165","31381059609"],["129746337890625","934173632812500","3082772988281250","6165545976562500","8323487068359375","7990547585625000","5593383309937500","2876597130825000","1078723924059375","287659713082500","51778748354850","5648590729620","282429536481"],["1946195068359375","15180321533203125","54649157519531250","120228146542968750","180342219814453125","194769597399609375","155815677919687500","93489406751812500","42070233038315625","14023411012771875","3365618643065250","550737596137950","55073759613795","2541865828329"],["29192926025390625","245220578613281250","956360256591796875","2295264615820312500","3787186616103515625","4544623939324218750","4090161545391796875","2804682202554375000","1472458156341046875","588983262536418750","176694978760925625","38551631729656500","5782744759448475","533791823949090","22876792454961"],["437893890380859375","3941045013427734375","16552389056396484375","43036211546630859375","77465180783935546875","102254038634794921875","102254038634794921875","78881686946841796875","47329012168105078125","22086872345115703125","7951274044241653125","2168529284793178125","433705856958635625","60051580194272625","5147278302366225","205891132094649"],["6568408355712890625","63056720214843750000","283755240966796875000","794514674707031250000","1549303615678710937500","2230997206577343750000","2454096927235078125000","2103511651915781250000","1419870365043152343750","757264194689681250000","318050961769666125000","104089405670072550000","26022351417518137500","4804126415541810000","617673396283947000","49413871702715760","1853020188851841"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224","63403380965376","1521681143169024","36520347436056576","876488338465357824","21035720123168587776","504857282956046106624","12116574790945106558976"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["225","270","81"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["15","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167564,"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-a617f0e181e2ec6f"]},{"id":167565,"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-a617f0e181e2ec6f"]},{"id":167566,"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-a617f0e181e2ec6f"]}],"region_exact":{},"incoming_edges":[{"id":"e-a617f0e181e2ec6f","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-3ccd0708169ccfae","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":"6b2647c75d079972c14f6927066b8220b51baebce91444ee0c02729fa98783f2","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-615df6ea54d3bbef","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-161da4a62428de73","kind":"fingerprint_related","node_a_id":"n-3ccd0708169ccfae","node_b_id":"n-384c17a53af3442c","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-384c17a53af3442c","other":{"id":"n-384c17a53af3442c","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","20"],["144","480","400"]],"card_rows":[["1"],["12","20"],["144","480","400"],["1728","8640","14400","8000"]],"row_sums_preview":["1","32","1024","32768","1048576","33554432","1073741824","34359738368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611"],"edges":["e-62a1b0a86f60f435"],"text":"A038271 \u2192 basis A013611"},"headline":"A038271 \u2192 basis A013611","chain":null}},{"id":"l-20ad2a147934c985","kind":"fingerprint_related","node_a_id":"n-c44f7114310405ad","node_b_id":"n-3ccd0708169ccfae","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-c44f7114310405ad","other":{"id":"n-c44f7114310405ad","name":"Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","5"],["9","30","25"]],"card_rows":[["1"],["3","5"],["9","30","25"],["27","135","225","125"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013622"],"derivation":{"root":"n-c44f7114310405ad","steps":[],"edges":[],"text":"A013622"},"headline":"A013622: Triangle of coefficients in expansion of (3+5x)^n.","chain":null}},{"id":"l-a94bb23f13aff319","kind":"fingerprint_related","node_a_id":"n-d524cbe046c3619e","node_b_id":"n-3ccd0708169ccfae","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-d524cbe046c3619e","other":{"id":"n-d524cbe046c3619e","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","120","100"],["216","1080","1800","1000"],["1296","8640","21600","24000","10000"]],"card_rows":[["36","120","100"],["216","1080","1800","1000"],["1296","8640","21600","24000","10000"],["7776","64800","216000","360000","300000","100000"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609"],"edges":["e-31086658f6c15e80"],"text":"A013612 \u2192 basis A013609"},"headline":"A013612 \u2192 basis A013609","chain":null}},{"id":"l-b391a75342293292","kind":"fingerprint_related","node_a_id":"n-05c39cd3844b146d","node_b_id":"n-3ccd0708169ccfae","fingerprint":"f6-b1029f17d1b64336","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1029f17d1b64336"},"other_id":"n-05c39cd3844b146d","other":{"id":"n-05c39cd3844b146d","name":"basis_map_A007318 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-b1029f17d1b64336","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","3"],["25","30","9"]],"card_rows":[["1"],["5","3"],["25","30","9"],["125","225","135","27"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A007318"],"edges":["e-66f5a589f0412003"],"text":"A013620 \u2192 basis A007318"},"headline":"A013620 \u2192 basis A007318","chain":null}},{"id":"l-45add3e5e3a318a9","kind":"oeis_match","node_a_id":"n-3ccd0708169ccfae","node_b_id":null,"fingerprint":null,"oeis_id":"A009968","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009968","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009968","relation":"match_row_sums"}],"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:           225,            270,              81;\n  1:          3375,           6075,            3645,             729;\n  2:         50625,         121500,          109350,           43740,            6561;\n  3:        759375,        2278125,         2733750,         1640250,          492075,           59049;\n  4:      11390625,       41006250,        61509375,        49207500,        22143375,         5314410,          531441;\n  5:     170859375,      717609375,      1291696875,      1291696875,       775018125,       279006525,        55801305,        4782969;\n  6:    2562890625,    12301875000,     25833937500,     31000725000,     23250543750,     11160261000,      3348078300,      573956280,       43046721;\n  7:   38443359375,   207594140625,    498225937500,    697516312500,    627764681250,    376658808750,    150663523500,    38742048900,     5811307335,     387420489;\n  8:  576650390625,  3459902343750,   9341736328125,  14946778125000,  15694117031250,  11299764262500,   5649882131250,  1937102445000,   435848050125,   58113073350,   3486784401;\n  9: 8649755859375, 57088388671875, 171265166015625, 308277298828125, 369932758593750, 310743517218750, 186446110331250, 79905475856250, 23971642756875, 4794328551375, 575319426165, 31381059609;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (15+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 270*x + 81*x^2.","mathematica":"P[0, x_] := 225 + 270*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[(15 + 9*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] = 225 + 270*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 9*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-615df6ea54d3bbef","steps":["basis A013610"],"edges":["e-a617f0e181e2ec6f"],"text":"A038329 \u2192 basis A013610"},"headline":"A038329 \u2192 basis A013610","chain":null}