{"id":"n-688fdf685007c036","revision":1,"first_index":0,"fingerprint":"f6-700950e0f523054a","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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":[["289","340","100"],["4913","8670","5100","1000"],["83521","196520","173400","68000","10000"],["1419857","4176050","4913000","2890000","850000","100000"],["24137569","85191420","125281500","98260000","43350000","10200000","1000000"],["410338673","1689629830","2981699700","2923235000","1719550000","606900000","119000000","10000000"],["6975757441","32827093840","67585193200","79511992000","58464700000","27512800000","8092000000","1360000000","100000000"],["118587876497","627818169690","1477219222800","2027555796000","1789019820000","1052364600000","412692000000","104040000000","15300000000","1000000000"],["2015993900449","11858787649700","31390908484500","49240640760000","50688894900000","35780396400000","17539410000000","5895600000000","1300500000000","170000000000","10000000000"],["34271896307633","221759329049390","652233320733500","1150999977765000","1354117620900000","1115155687800000","655973934000000","275619300000000","81064500000000","15895000000000","1870000000000","100000000000"],["582622237229761","4112627556915960","13305559742963400","26089332829340000","34529999332950000","32498822901600000","22303113756000000","11245267440000000","4134289500000000","1080860000000000","190740000000000","20400000000000","1000000000000"],["9904578032905937","75740890839868930","267320791199537400","576574255528414000","847903316953550000","897779982656700000","704141162868000000","414200684040000000","182735595900000000","59717515000000000","14051180000000000","2254200000000000","221000000000000","10000000000000"],["168377826559400929","1386640924606831180","5301862358790825100","12474970255978412000","20180098943494490000","23741292874699400000","20948199595323000000","14082823257360000000","7248511970700000000","2842553714000000000","836045210000000000","178833200000000000","26299000000000000","2380000000000000","100000000000000"],["2862423051509815793","25256673983910139350","103998069345512338500","265093117939541255000","467811384599190450000","605402968304834700000","593532321867485000000","448889991328350000000","264052936075500000000","120808532845000000000","42638305710000000000","11400616500000000000","2235415000000000000","303450000000000000","25500000000000000","1000000000000000"],["48661191875666868481","457987688241570526880","2020533918712811148000","5546563698427324720000","10603724717581650200000","14969964307174094400000","16144079154795592000000","13566453071256800000000","8977799826567000000000","4694274419120000000000","1932936525520000000000","620193537600000000000","152008220000000000000","27512800000000000000","3468000000000000000","272000000000000000","10000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987","205891132094649","5559060566555523","150094635296999121","4052555153018976267","109418989131512359209","2954312706550833698643","79766443076872509863361"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["289","340","100"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["17","10"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166881,"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-2f341f062a4ee4be"]},{"id":166882,"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-2f341f062a4ee4be"]},{"id":166883,"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-2f341f062a4ee4be"]}],"region_exact":{},"incoming_edges":[{"id":"e-2f341f062a4ee4be","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-688fdf685007c036","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":"c78df164906b832527573d6c2ed48fa279df5dc8b4e89895099046a4f462b2fd","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-a5302200d285579d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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-b6dc9defe538c975","kind":"oeis_match","node_a_id":"n-688fdf685007c036","node_b_id":null,"fingerprint":null,"oeis_id":"A009971","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":"A009971","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009971","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:            289,             340,             100;\n  1:           4913,            8670,            5100,             1000;\n  2:          83521,          196520,          173400,            68000,            10000;\n  3:        1419857,         4176050,         4913000,          2890000,           850000,           100000;\n  4:       24137569,        85191420,       125281500,         98260000,         43350000,         10200000,         1000000;\n  5:      410338673,      1689629830,      2981699700,       2923235000,       1719550000,        606900000,       119000000,        10000000;\n  6:     6975757441,     32827093840,     67585193200,      79511992000,      58464700000,      27512800000,      8092000000,      1360000000,      100000000;\n  7:   118587876497,    627818169690,   1477219222800,    2027555796000,    1789019820000,    1052364600000,    412692000000,    104040000000,    15300000000,     1000000000;\n  8:  2015993900449,  11858787649700,  31390908484500,   49240640760000,   50688894900000,   35780396400000,  17539410000000,   5895600000000,  1300500000000,   170000000000,   10000000000;\n  9: 34271896307633, 221759329049390, 652233320733500, 1150999977765000, 1354117620900000, 1115155687800000, 655973934000000, 275619300000000, 81064500000000, 15895000000000, 1870000000000, 100000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (17+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 340*x + 100*x^2.","mathematica":"P[0, x_] := 289 + 340*x + 100*x^2;\nP[n_, x_] := P[n, x] = Expand[(17 + 10*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] = 289 + 340*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 10*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-a5302200d285579d","steps":["basis A013609"],"edges":["e-2f341f062a4ee4be"],"text":"A038331 \u2192 basis A013609"},"headline":"A038331 \u2192 basis A013609","chain":null}
