{"id":"n-784296607d60c8c4","revision":0,"first_index":0,"fingerprint":"f6-b9f56a026bd52dff","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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":[["697","696","144"],["15317","25092","12528","1728"],["319345","735216","602208","200448","20736"],["6431117","19160700","22056480","12044160","3006720","248832"],["126367273","463040424","689785200","529355520","216794880","43296768","2985984"],["2437894469","10614850932","19447697808","19313985600","11116465920","3642153984","606154752","35831808"],["46368270049","234037869024","509512844736","622326329856","463535654400","213436145664","58274463744","8312979456","429981696"],["871969680125","5007773165292","12638044927296","18342462410496","16802810906112","10012370135040","3841850621952","899091726336","112225222656","5159780352"],["16246539080089","104636361615000","300466389917520","505521797091840","550273872314880","403267461746688","200247402700800","65860296376320","13486375895040","1496336302080","61917364224"],["300383091166901","2144543158571748","6905999866590000","13220521156370880","16682219304030720","14527230229112832","8871884158427136","3776093879500800","1086694890209280","197800179793920","19751639187456","743008370688"],["5517775305528913","43255165128033744","154407107417165856","331487993596320000","475938761629351680","480447915956084736","348653525498707968","182507331259072512","67969689831014400","17387118243348480","2848322589032448","258566912999424","8916100448256"],["100793647040748653","860772947662510428","3373902879986632032","8029169585692624512","12928031750256480000","14849289362835772416","12491645814858203136","7769992853971206144","3558892959551913984","1178141290404249600","271239044596236288","40394393080823808","3361369868992512","106993205379072"],["1832346936087598345","16933332702845773704","72304927603650875952","188938561279251393792","337225122599090229504","434381866808617728000","415780102159401627648","299799499556596875264","163169849933395329024","66432668578302394368","19792773678791393280","4142559953833426944","565521503131533312","43439241383903232","1283918464548864"],["33170431832201983013","329822448495767702100","1523999943256119633360","4338295656219052557120","8502235257566312720640","12140104413567248262144","13031456004258531840000","10691488341241756139520","6745488740023429693440","3263396998667906580480","1195788034409443098624","323881751107495526400","62138399307501404160","7830297735667384320","558504532078755840","15407021574586368"],["598246417765551500737","6368722911782780738496","31662955055593699401600","97535996368391656535040","208238191498514522741760","326485833890546408472576","388483341234151944388608","357434221831091159040000","256595720189802147348480","143903759787166500126720","62657222374423806345216","20871936600601188630528","5182108017719928422400","917736359003097661440","107386940374866984960","7148858010608074752","184884258895036416"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1537","54665","1877953","62948009","2071626049","67215035273","2156234538625","68533758575657","2161564750404481","67733864561332361","2110688189467108417","65455734893011772585","2021343856452844631233","62189670629242087867529","1907050713219198761449729"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["697","696","144"]],"recurrence":{"denominator":{"coeffs":[["29","12"],["12","0"]]},"terms":[{"coeff":{"coeffs":[["697","696","144"],["204","144","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194293,"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-653374463eddd60b"]},{"id":194294,"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-653374463eddd60b"]},{"id":194295,"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-653374463eddd60b"]},{"id":194296,"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-653374463eddd60b"]}],"region_exact":{},"incoming_edges":[{"id":"e-653374463eddd60b","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-784296607d60c8c4","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":"fcdf44013b4e0417c3df3497a690fe68ce6ccfbdce7efb3c990fc7f5fdef5dc3","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-b69b5201d4a2fd8f","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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:             697,              696,              144;\n  1:           15317,            25092,            12528,              1728;\n  2:          319345,           735216,           602208,            200448,             20736;\n  3:         6431117,         19160700,         22056480,          12044160,           3006720,            248832;\n  4:       126367273,        463040424,        689785200,         529355520,         216794880,          43296768,          2985984;\n  5:      2437894469,      10614850932,      19447697808,       19313985600,       11116465920,        3642153984,        606154752,         35831808;\n  6:     46368270049,     234037869024,     509512844736,      622326329856,      463535654400,      213436145664,      58274463744,       8312979456,        429981696;\n  7:    871969680125,    5007773165292,   12638044927296,    18342462410496,    16802810906112,    10012370135040,    3841850621952,     899091726336,     112225222656,      5159780352;\n  8:  16246539080089,  104636361615000,  300466389917520,   505521797091840,   550273872314880,   403267461746688,  200247402700800,   65860296376320,   13486375895040,   1496336302080,    61917364224;\n  9: 300383091166901, 2144543158571748, 6905999866590000, 13220521156370880, 16682219304030720, 14527230229112832, 8871884158427136, 3776093879500800, 1086694890209280, 197800179793920, 19751639187456, 743008370688;\n","recurrence_sentence":"The row polynomials satisfy (12*n+29+12*x)*P(n,x) = (204*n+697+(144*n+696)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 697 + 696*x + 144*x^2.","mathematica":"P[0, x_] := 697 + 696*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((204*n + 697 + (144*n + 696)*x + 144*x^2)*P[n - 1, x])/(12*n + 29 + 12*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 697 + 696*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((204*n + 697 + (144*n + 696)*x + 144*x^2)*v[i - 1])/(12*n + 29 + 12*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-1f2dec04776afb2b","steps":["basis A013609","D+1"],"edges":["e-7304be3f66cc3caf","e-653374463eddd60b"],"text":"A038320 \u2192 basis A013609 \u2192 D+1"},"headline":"A038320 \u2192 basis A013609 \u2192 D+1","chain":null}