{"id":"n-d34ea40cdf30ed64","revision":0,"first_index":0,"fingerprint":"f6-a55687d918dbb23b","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(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":[["1425","2632","784"],["37183","119700","110544","21952"],["898529","4164496","6703200","4126976","614656"],["20721039","125794060","291514720","312816000","144444160","17210368"],["463030513","3481134552","10566701040","16324824320","13138272000","4853323776","481890304"],["10114864415","90753980548","341151186096","690357801280","799916391680","515020262400","158541910016","13492928512"],["217210832577","2265729628960","10164445821376","25472621895168","38660036871680","35836254347264","19227423129600","5073341120512","377801998336"],["4602545584111","54737129809404","285481933248960","853813448995584","1604775179395584","1948465858332672","1505122682585088","692187232665600","159810245296128","10578455953408"],["96483621635921","1288712763551080","7663198173316560","26644980436569600","59766941429690880","89867410046152704","90928406722191360","60204907303403520","24226553143296000","4971874298101760","296196766695424"],["2004858666300927","29716955463863668","198461765586866320","786755012460500160","2051663493615859200","3681643592068958208","4613193715702505472","4000849895776419840","2317888931181035520","829086485348352000","153133728381534208","8293509467471872"],["41354041908867745","673632511877111472","4992448517929096224","22227717745729027840","66087421046682013440","137871786770985738240","206172041155861659648","221433298353720262656","168035695622609633280","86534520097425326080","27857305907704627200","4677539339654135808","232218265089212416"],["847699614002339663","15052871254827859180","122601117161634287904","605750420175397008512","2022722314861341533440","4811164252198450578432","8364221730773134786560","10720946140104806301696","10075215075094271950848","6796110356292211834880","3149856531546281869312","921823577309498572800","141885359969508786176","6502111422497947648"],["17283776202987651249","332298248688917147896","2950362765946260399280","16019879309120213619456","59363541177188906834176","158581429485129176221696","314329397810298771124224","468396416923295548047360","525326360865135508783104","438831589937439400525824","266407525966654703927296","112249432760558408433664","30112903525443620044800","4278389316003649552384","182059119829942534144"],["350763935058686129119","7259186005254813524580","69782632224672601058160","413050787232476455899200","1682087327457622430042880","4986537458883868174070784","11100700063959042335518720","18859763868617926267453440","24590811888473016272486400","24515230173706323743211520","18430926777372454822084608","10171923718726815968133120","3928730146619544295178240","972878421591255416832000","128351679480109486571520","5097655355238390956032"],["7089586322951530805633","157142242906291385845312","1626057665177078229505920","10420873078884441758018560","46261688170037363060710400","150715024540202969731842048","372328130263328823663951872","710444804093378709473198080","1056146776642603870977392640","1224075969559545698897100800","1098282311782043303695876096","750641381478441796390354944","379751818832467796143636480","135390085052735064941527040","31132109490920173338624000","3833436827139269998936064","142734349946674946768896"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["4841","289379","16507857","912500347","49309176505","2619349324947","137294865645473","7119574711866539","365955664704604809","18672691943719667395","946836979377176712049","47754699557777865657051","2397378870986936014846553","119863482549640680159150323","5971357433656082278164094785"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1425","2632","784"]],"recurrence":{"denominator":{"coeffs":[["47","28"],["28","0"]]},"terms":[{"coeff":{"coeffs":[["1425","2632","784"],["532","784","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195035,"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-e731feceb189abfc"]},{"id":195036,"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-e731feceb189abfc"]},{"id":195037,"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-e731feceb189abfc"]},{"id":195038,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-e731feceb189abfc"]}],"region_exact":{},"incoming_edges":[{"id":"e-e731feceb189abfc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-d34ea40cdf30ed64","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"4788ce67bd998872cec3c07ce089c333f0785ddfe3f2e3af8330d5dcc3ce5222","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3afe7d5f29fd0242","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(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":[],"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:             1425,              2632,                784;\n  1:            37183,            119700,             110544,              21952;\n  2:           898529,           4164496,            6703200,            4126976,              614656;\n  3:         20721039,         125794060,          291514720,          312816000,           144444160,            17210368;\n  4:        463030513,        3481134552,        10566701040,        16324824320,         13138272000,          4853323776,           481890304;\n  5:      10114864415,       90753980548,       341151186096,       690357801280,        799916391680,        515020262400,        158541910016,         13492928512;\n  6:     217210832577,     2265729628960,     10164445821376,     25472621895168,      38660036871680,      35836254347264,      19227423129600,       5073341120512,        377801998336;\n  7:    4602545584111,    54737129809404,    285481933248960,    853813448995584,    1604775179395584,    1948465858332672,    1505122682585088,     692187232665600,     159810245296128,     10578455953408;\n  8:   96483621635921,  1288712763551080,   7663198173316560,  26644980436569600,   59766941429690880,   89867410046152704,   90928406722191360,   60204907303403520,   24226553143296000,   4971874298101760,    296196766695424;\n  9: 2004858666300927, 29716955463863668, 198461765586866320, 786755012460500160, 2051663493615859200, 3681643592068958208, 4613193715702505472, 4000849895776419840, 2317888931181035520, 829086485348352000, 153133728381534208, 8293509467471872;\n","recurrence_sentence":"The row polynomials satisfy (28*n+47+28*x)*P(n,x) = (532*n+1425+(784*n+2632)*x+784*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1425 + 2632*x + 784*x^2.","mathematica":"P[0, x_] := 1425 + 2632*x + 784*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((532*n + 1425 + (784*n + 2632)*x + 784*x^2)*P[n - 1, x])/(28*n + 47 + 28*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1425 + 2632*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((532*n + 1425 + (784*n + 2632)*x + 784*x^2)*v[i - 1])/(28*n + 47 + 28*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-5cf6acfdac607d6d","steps":["basis A013611","D+1"],"edges":["e-0d83dd77e9f6bf78","e-e731feceb189abfc"],"text":"A038333 \u2192 basis A013611 \u2192 D+1"},"headline":"A038333 \u2192 basis A013611 \u2192 D+1","chain":null}