{"id":"n-89e4696a36e5da90","revision":0,"first_index":0,"fingerprint":"f6-8f1c79a7a65ded7e","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["41","20"],["1681","1640","400"],["68921","100860","49200","8000"],["2825761","5513680","4034400","1312000","160000"],["115856201","282576100","275684000","134480000","32800000","3200000"],["4750104241","13902744120","16954566000","11027360000","4034400000","787200000","64000000"],["194754273881","665014593740","973192088400","791213080000","385957600000","112963200000","18368000000","1280000000"],["7984925229121","31160683820960","53201167499200","51903578048000","31648523200000","12350643200000","3012352000000","419840000000","25600000000"],["327381934393961","1437286541241780","2804461543886400","3192070049952000","2335661012160000","1139346835200000","370519296000000","77460480000000","9446400000000","512000000000"],["13422659310152401","65476386878792200","143728654124178000","186964102925760000","159603502497600000","93426440486400000","37978227840000000","10586265600000000","1936512000000000","209920000000000","10240000000000"],["550329031716248441","2952985048233528220","7202402556667142000","10540101302439720000","10283025660916800000","7022554109894400000","3425636151168000000","1193601446400000000","291122304000000000","47336960000000000","4618240000000000","204800000000000"],["22563490300366186081","132078967611899625840","354358205788023386400","576192204533371360000","632406078146383200000","493585231724006400000","280902164395776000000","117450382325760000000","35808043392000000000","7763261440000000000","1136087040000000000","100761600000000000","4096000000000000"],["925103102315013629321","5866507478095208381060","17170265789546951359200","30711044501628693488000","37452493294669138400000","32885116063611926400000","21388693374706944000000","10433508963271680000000","3817137425587200000000","1034454586880000000000","201844797440000000000","26852966400000000000","2183168000000000000","81920000000000000"],["37929227194915558802161","259028868648203816209880","821311046933329173348400","1602558140357715460192000","2149773115114008544160000","2097339624501471750400000","1534638749635223232000000","855547734988277760000000","365172813714508800000000","118755386573824000000000","28964728432640000000000","5137867571200000000000","626569216000000000000","47022080000000000000","1638400000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","61","3721","226981","13845841","844596301","51520374361","3142742836021","191707312997281","11694146092834141","713342911662882601","43513917611435838661","2654348974297586158321","161915287432152755657581","9876832533361318095112441"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["41","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193156,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-485f849784a4cddf"]},{"id":193157,"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-485f849784a4cddf","e-7113ade089e08d18"]},{"id":193158,"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-485f849784a4cddf","e-7113ade089e08d18"]},{"id":193159,"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-485f849784a4cddf"]},{"id":193160,"position":4,"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-7113ade089e08d18"]}],"region_exact":{},"incoming_edges":[{"id":"e-7113ade089e08d18","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-89e4696a36e5da90","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":"d88f79b4b0168d890235d9089efdb9fe47fbf7323d453a8f3e1a2bc0d8abe730","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-fbf962df53148c52","role":"sequence","name":"basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^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]"}]},{"id":"e-485f849784a4cddf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-89e4696a36e5da90","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"a7def260b57811c9afb5a027ff4244f678d7dae4dc5eec435e094477788d7de9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fbf962df53148c52","role":"sequence","name":"basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^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:               1;\n  1:              41,               20;\n  2:            1681,             1640,              400;\n  3:           68921,           100860,            49200,             8000;\n  4:         2825761,          5513680,          4034400,          1312000,           160000;\n  5:       115856201,        282576100,        275684000,        134480000,         32800000,          3200000;\n  6:      4750104241,      13902744120,      16954566000,      11027360000,       4034400000,        787200000,        64000000;\n  7:    194754273881,     665014593740,     973192088400,     791213080000,     385957600000,     112963200000,     18368000000,     1280000000;\n  8:   7984925229121,   31160683820960,   53201167499200,   51903578048000,   31648523200000,   12350643200000,   3012352000000,   419840000000,   25600000000;\n  9: 327381934393961, 1437286541241780, 2804461543886400, 3192070049952000, 2335661012160000, 1139346835200000, 370519296000000, 77460480000000, 9446400000000, 512000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (41+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(41 + 20*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (41 + 20*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-e1d1088f7d1ff894","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-25d25e50279a57da","e-485f849784a4cddf"],"text":"A038324 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038324 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}
