{"id":"n-42e457d54fd69d5d","revision":0,"first_index":0,"fingerprint":"f6-f2fb5cf749ee837b","name":"basis_map_A013609 of basis_map_A013611 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":[["1681","3936","2304"],["68921","242064","283392","110592"],["2825761","13232832","23238144","18137088","5308416"],["115856201","678182640","1587939840","1859051520","1088225280","254803968"],["4750104241","33366585888","97658300160","152442224640","133851709440","62681776128","12230590464"],["194754273881","1596035024976","5605586429184","10937729617920","12805146869760","8994834874368","3510179463168","587068342272"],["7984925229121","74785641170304","306438724795392","717515062935552","1050022043320320","983435279597568","575669431959552","192558416265216","28179280429056"],["327381934393961","3449487698980272","16153698492785664","44127176370536448","77491626797039616","90721904542875648","70807340131024896","35527027800932352","10398154478321664","1352605460594688"],["13422659310152401","157143328509101280","827877047755265280","2584591758845706240","5295261164464373760","7439196172515803136","7257752363430051840","4855360466127421440","2131621668055941120","554568238843822080","64925062108545024"],["550329031716248441","7087164115760467728","41485838726402737920","145706360404926689280","341166112167633223680","559179578967437869056","654649263181390675968","547441892555866767360","320453790764409815040","125055137859281879040","29281203010953805824","3116402981210161152"],["22563490300366186081","316989522268559102016","2041103265339014705664","7965281035469325680640","20981715898309443256320","39302336121711347367936","53681239580874035429376","53868282227497289908224","39415816264022407249920","20509042608922228162560","7203175940694636232704","1533270266755399286784","149587343098087735296"],["925103102315013629321","14079617947428500114544","98900730947790439828992","424549479190515058778112","1242583841533214806179840","2618518144109018518388736","4087442956657980126265344","4785299071209342586847232","4201726013744788612841472","2732829927638886902661120","1279764258796747037343744","408616526090313909927936","79730053871280762912768","7180192468708211294208"],["37929227194915558802161","621669284755689158903712","4730751630335976038486784","22153763732305058521694208","71324312504006529874722816","167003268302064069950570496","293274032140210074059538432","392394523839166092121473024","401965121981584777295167488","313728875692944216425496576","183646171137333199858827264","78181961991946728099545088","22882525461057578955964416","4121430477038513282875392","344649238497994142121984"],["1555098314991537910888601","27309043580339202337555920","223800942512048097205336320","1135380391280634249236828160","3987677471814910533904957440","10270701000576940301960085504","20040392196247688394068459520","30165329020135893331838238720","35315507145524948290932572160","32157209758526782183613399040","22588479049891983582635753472","12020476656261809445305057280","4690917719516803685972705280","1267339871689342834484183040","211959281676266397405020160","16543163447903718821855232"],["63759030914653054346432641","1194315505913501115562445568","10486672734850253697621473280","57293041283084312884566097920","217993035125881775853471006720","612507259670770258007801462784","1314649728073848358650890944512","2198717315245460669520653844480","2895871585933045759856470917120","3013589943084795587492912824320","2469673709454856871701509046272","1577086537301549399224023515136","769310506000755804499523665920","277124985276069633140541358080","69522644389815378348846612480","10852315221824839547137032192","794071845499378503449051136"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["7921","704969","62742241","5584059449","496981290961","44231334895529","3936588805702081","350356403707485209","31181719929966183601","2775173073766990340489","246990403565262140303521","21982145917308330487013369","1956410986640441413344189841","174120577810999285787632895849","15496731425178936435099327730561"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1681","3936","2304"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["41","48"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201201,"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-69d18771ea5e0a84"]},{"id":201202,"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-69d18771ea5e0a84"]},{"id":201203,"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-69d18771ea5e0a84"]}],"region_exact":{},"incoming_edges":[{"id":"e-69d18771ea5e0a84","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-42e457d54fd69d5d","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":"b4f47066449546c3a70f37f083ecb96270b5b6e317d29b9293f81fa2a86b5c8e","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-2117bc35fff39734","role":"sequence","name":"basis_map_A013611 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:               1681,                3936,                 2304;\n  1:              68921,              242064,               283392,                110592;\n  2:            2825761,            13232832,             23238144,              18137088,               5308416;\n  3:          115856201,           678182640,           1587939840,            1859051520,            1088225280,             254803968;\n  4:         4750104241,         33366585888,          97658300160,          152442224640,          133851709440,           62681776128,           12230590464;\n  5:       194754273881,       1596035024976,        5605586429184,        10937729617920,        12805146869760,         8994834874368,         3510179463168,          587068342272;\n  6:      7984925229121,      74785641170304,      306438724795392,       717515062935552,      1050022043320320,       983435279597568,       575669431959552,       192558416265216,        28179280429056;\n  7:    327381934393961,    3449487698980272,    16153698492785664,     44127176370536448,     77491626797039616,     90721904542875648,     70807340131024896,     35527027800932352,     10398154478321664,      1352605460594688;\n  8:  13422659310152401,  157143328509101280,   827877047755265280,   2584591758845706240,   5295261164464373760,   7439196172515803136,   7257752363430051840,   4855360466127421440,   2131621668055941120,    554568238843822080,    64925062108545024;\n  9: 550329031716248441, 7087164115760467728, 41485838726402737920, 145706360404926689280, 341166112167633223680, 559179578967437869056, 654649263181390675968, 547441892555866767360, 320453790764409815040, 125055137859281879040, 29281203010953805824, 3116402981210161152;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (41+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1681 + 3936*x + 2304*x^2.","mathematica":"P[0, x_] := 1681 + 3936*x + 2304*x^2;\nP[n_, x_] := P[n, x] = Expand[(41 + 48*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] = 1681 + 3936*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (41 + 48*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-1f2dec04776afb2b","steps":["basis A013611","basis A013609"],"edges":["e-5fcdd9bd1fb647dd","e-69d18771ea5e0a84"],"text":"A038320 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038320 \u2192 basis A013611 \u2192 basis A013609","chain":null}