{"id":"n-ba23d827179717e1","revision":1,"first_index":0,"fingerprint":"f6-751f1aae2d0ee2d6","name":"basis_map_A008277 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":false,"rows":[["289","195","25"],["4913","5735","1650","125"],["83521","150735","73225","12250","625"],["1419857","3733775","2730750","758125","84375","3125"],["24137569","89242335","92399125","37913750","6912500","553125","15625"],["410338673","2084019215","2940988050","1675235625","445331250","57793750","3500000","78125"],["6975757441","47900116095","89826773425","68312480250","24853434375","4653993750","453468750","21562500","390625"],["118587876497","1088681341295","2664823462950","2635133235125","1261139473125","319734909375","44583000000","3388593750","130078125","1953125"],["2015993900449","24543928890975","77393640206125","97648380838750","59837826681250","19734563559375","3694075546875","399121875000","24357421875","771484375","9765625"],["34271896307633","550046405103695","2212347930020250","3511716387870625","2702241491400000","1128040802900000","272294368500000","39224715234375","3383982421875","169619140625","4511718750","48828125"],["582622237229761","12272380393819455","62483626136065225","123436664061961250","117541517121153125","60888921178800000","18438039334000000","3401157034687500","389010574218750","27436298828125","1150380859375","26074218750","244140625"],["9904578032905937","272905479850176815","1748419807642858350","4262391380663086125","4966219453792471875","3145042275115365625","1171032454592000000","269050362473750000","39179387903906250","3646103398437500","214257011718750","7629248046875","149169921875","1220703125"],["168377826559400929","6053443446868419615","48571862205608059525","145138623219433047750","205062076693636890000","156922872823807715625","70763736741400828125","19845781121595000000","3578476922891406250","421955350222656250","32585736777343750","1620590917968750","49632324218750","845947265625","6103515625"],["2862423051509815793","134017644963902236175","1341707496785759705250","4887295254049897825625","8312989953761730168750","7616071042068108506250","4110509990964877500000","1385799302029944140625","303202090212785156250","44053616328261718750","4293021115195312500","279611229980468750","11924643554687500","317529296875000","4760742187500","30517578125"],["48661191875666868481","2962700304463398274815","36896190638035023222625","163101985613525528946250","332017104559433505371875","361439933535669208106250","231274324785689785031250","92614113660381482812500","24211515652278474609375","4247334663416152343750","507900496359394531250","41597114134570312500","2316253703613281250","85660620117187500","2001831054687500","26611328125000","152587890625"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["509","12423","320356","8730007","251174029","7617284688","242997977211","8136203923367","285292670170674","10453742159445953","399462551069981941","15888013320115349102","656561463164093736569","28143093693526063882843","1249365766158180891602796"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["289","195","25"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["17","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166169,"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-fd4b406e93b80ec2"]},{"id":166170,"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-fd4b406e93b80ec2"]},{"id":166171,"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-fd4b406e93b80ec2"]}],"region_exact":{},"incoming_edges":[{"id":"e-fd4b406e93b80ec2","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-ba23d827179717e1","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"5f32f2d18fc7f095b890d0988c0cefcb362bf861aa28c5acbe72c47541d4ce58","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":[],"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: 289, 195, 25;\n 1: 4913, 5735, 1650, 125;\n 2: 83521, 150735, 73225, 12250, 625;\n 3: 1419857, 3733775, 2730750, 758125, 84375, 3125;\n 4: 24137569, 89242335, 92399125, 37913750, 6912500, 553125, 15625;\n 5: 410338673, 2084019215, 2940988050, 1675235625, 445331250, 57793750, 3500000, 78125;\n 6: 6975757441, 47900116095, 89826773425, 68312480250, 24853434375, 4653993750, 453468750, 21562500, 390625;\n 7: 118587876497, 1088681341295, 2664823462950, 2635133235125, 1261139473125, 319734909375, 44583000000, 3388593750, 130078125, 1953125;\n 8: 2015993900449, 24543928890975, 77393640206125, 97648380838750, 59837826681250, 19734563559375, 3694075546875, 399121875000, 24357421875, 771484375, 9765625;\n 9: 34271896307633, 550046405103695, 2212347930020250, 3511716387870625, 2702241491400000, 1128040802900000, 272294368500000, 39224715234375, 3383982421875, 169619140625, 4511718750, 48828125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (17+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 289 + 195*x + 25*x^2.","mathematica":"P[0, x_] := 289 + 195*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(17 + 5*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 195*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 5*x)*v[i - 1] + 5*x*deriv(v[i - 1], 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-a5302200d285579d","steps":["basis A008277"],"edges":["e-fd4b406e93b80ec2"],"text":"A038331 \u2192 basis A008277"},"headline":"A038331 \u2192 basis A008277","chain":null}