{"id":"n-2b8945493c55cc3d","revision":1,"first_index":0,"fingerprint":"f6-da9c969e22754650","name":"theta_plus_one of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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":[["2"],["2","8"],["2","24","24"],["2","56","144","64"],["2","120","600","640","160"],["2","248","2160","4160","2400","384"],["2","504","7224","22400","22400","8064","896"],["2","1016","23184","108864","168000","102144","25088","2048"],["2","2040","72600","497280","1112160","1016064","413952","73728","4608"],["2","4088","223920","2182720","6804000","8765568","5268480","1536000","207360","10240"],["2","8184","684024","9328000","39476800","68923008","57332352","24330240","5322240","563200","22528"],["2","16376","2076624","39136064","220704000","508282368","562146816","325687296","102643200","17459200","1486848","49152"],["2","32760","6279000","162081920","1201360160","3579383808","5121019904","3890405376","1656585216","402688000","54765568","3833856","106496"],["2","65528","18935280","665071680","6412005600","24359567232","44199034880","42828431360","23662679040","7708180480","1488334848","165494784","9691136","229376"],["2","131064","57002424","2710780800","33722707200","161546216832","366232234368","443653816320","309328081920","129665536000","33329680384","5233213440","484556800","24084480","491520"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","10","50","266","1522","9354","61490","430346","3192434","25002378","205990578","1779687946","16078542066","151517721226","1485988533554"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["2","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165900,"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-36c887a5cdb807b7"]},{"id":165901,"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-36c887a5cdb807b7"]},{"id":165902,"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-36c887a5cdb807b7"]},{"id":165903,"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-36c887a5cdb807b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-36c887a5cdb807b7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2b8945493c55cc3d","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":"201ff59908b83890b9c70e240142d8a9a0c80980bd6f2b89cb1639a33ec2cd7c","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-d5fb64ec7bd8ef70","role":"sequence","name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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: 2;\n  1: 2,    8;\n  2: 2,   24,     24;\n  3: 2,   56,    144,      64;\n  4: 2,  120,    600,     640,     160;\n  5: 2,  248,   2160,    4160,    2400,     384;\n  6: 2,  504,   7224,   22400,   22400,    8064,     896;\n  7: 2, 1016,  23184,  108864,  168000,  102144,   25088,    2048;\n  8: 2, 2040,  72600,  497280, 1112160, 1016064,  413952,   73728,   4608;\n  9: 2, 4088, 223920, 2182720, 6804000, 8765568, 5268480, 1536000, 207360, 10240;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -4*x^2*P(n-2,x) - 2*x^2*d/dx P(n-2,x) + (1+4*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-4*x^2*P[n - 2, x] - 2*x^2*D[P[n - 2, x], x] + (1 + 4*x)*P[n - 1, x] + 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] = 2; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -4*x^2*v[i - 2] - 2*x^2*deriv(v[i - 2], x) + (1 + 4*x)*v[i - 1] + 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-d5fb64ec7bd8ef70","steps":["\u03b8+1"],"edges":["e-36c887a5cdb807b7"],"text":"A227450 \u2192 \u03b8+1"},"headline":"A227450 \u2192 \u03b8+1","chain":null}