{"id":"n-52d53da180a91748","revision":0,"first_index":0,"fingerprint":"f6-bab60f2685452b19","name":"theta_plus_one_squared of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^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":false,"rows":[["1"],["9","16"],["81","288","144"],["729","3888","3888","1024"],["6561","46656","69984","36864","6400"],["59049","524880","1049760","829440","288000","36864"],["531441","5668704","14171760","14929920","7776000","1990656","200704"],["4782969","59521392","178564176","235146240","163296000","62705664","12644352","1048576"],["43046721","612220032","2142770112","3386105856","2939328000","1504935936","455196672","75497472","5308416"],["387420489","6198727824","24794911296","45712429056","47617113600","30474952704","12290310144","3057647616","429981696","26214400"],["3486784401","61987278240","278942752080","587731230720","714256704000","548549148672","276531978240","91729428480","19349176320","2359296000","126877696"],["31381059609","613674054576","3068370272880","7273173980160","10101630528000","9051060953088","5475333169152","2270303354880","638522818560","116785152000","12560891904","603979776"],["282429536481","6025163444928","33138398947104","87278087761920","136372012128000","139644940419072","98555997044736","49038552465408","17240116101120","4204265472000","678288162816","65229815808","2835349504"],["2541865828329","58745343588048","352472061528288","1021153626814464","1772836157664000","2042307253628928","1647293093462016","956251773075456","403418716766208","122974765056000","26453238349824","3815944224768","331735891968","13153337344"],["22876792454961","569377945545696","3700956646047024","11696850634420224","22337735586566400","28592301550804992","25944866222026752","17212531915358208","8471793052090368","3098964079411200","833277008019456","160269657440256","20899361193984","1657320505344","60397977600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","25","513","9529","166465","2787993","45269185","717709369","11164409217","170989708825","2585050654849","38653400214585","572526316648897","8410608729215641","122664418169862465"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["81","-36","0"],["0","108","0"],["0","0","16"]]},"terms":[{"coeff":{"coeffs":[["729","972","432","64"],["0","972","720","128"],["0","0","144","64"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187423,"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-e6f1072c572d1f16"]},{"id":187424,"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-e6f1072c572d1f16"]},{"id":187425,"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-e6f1072c572d1f16"]},{"id":187426,"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-e6f1072c572d1f16"]}],"region_exact":{},"incoming_edges":[{"id":"e-e6f1072c572d1f16","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-52d53da180a91748","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"c78b077879390a8b5c457da03196133318ea0bc64be02a47b555458599ed42d7","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-b8c0cc94a00a4dd8","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^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:         9,         16;\n  2:        81,        288,         144;\n  3:       729,       3888,        3888,        1024;\n  4:      6561,      46656,       69984,       36864,        6400;\n  5:     59049,     524880,     1049760,      829440,      288000,       36864;\n  6:    531441,    5668704,    14171760,    14929920,     7776000,     1990656,      200704;\n  7:   4782969,   59521392,   178564176,   235146240,   163296000,    62705664,    12644352,    1048576;\n  8:  43046721,  612220032,  2142770112,  3386105856,  2939328000,  1504935936,   455196672,   75497472,   5308416;\n  9: 387420489, 6198727824, 24794911296, 45712429056, 47617113600, 30474952704, 12290310144, 3057647616, 429981696, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (81+(108*n-36)*x+16*n^2*x^2)*P(n,x) = (729+(972*n+972)*x+(144*n^2+720*n+432)*x^2+(64*n^2+128*n+64)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((729 + (972*n + 972)*x + (144*n^2 + 720*n + 432)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(81 + (108*n - 36)*x + 16*n^2*x^2)]];\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] = ((729 + (972*n + 972)*x + (144*n^2 + 720*n + 432)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(81 + (108*n - 36)*x + 16*n^2*x^2)); 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-366b2525683554ec","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-b30dba0e9f8b339c","e-e6f1072c572d1f16"],"text":"A038279 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038279 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}