{"id":"n-c05cf5b45147fec7","revision":0,"first_index":1,"fingerprint":"f6-692d0ae5c5a32a15","name":"theta_plus_one of Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","2"],["60","40","3"],["840","840","126","4"],["15120","20160","4536","288","5"],["332640","554400","166320","15840","550","6"],["8648640","17297280","6486480","823680","42900","936","7"],["259459200","605404800","272432160","43243200","3003000","98280","1470","8"],["8821612800","23524300800","12350257920","2352430080","204204000","8910720","199920","2176","9"],["335221286400","1005663859200","603398315520","134088514560","13967553600","761866560","22790880","372096","3078","10"],["14079294028800","46930980096000","31678411564800","8045310873600","977728752000","63996791040","2393042400","52093440","646380","4200","11"],["647647525324800","2374707592857600","1781030694643200","508865912755200","70675821216000","5397062711040","242175890880","6589820160","109022760","1062600","5566","12"],["32382376266240000","129529505064960000","106861841678592000","33924394183680000","5300686591200000","462605375232000","24217589088000","790778419200","16353414000","212520000","1669800","7200","13"],["1748648318376960000","7577476046300160000","6819728441670144000","2381492471694336000","413453554113600000","40593621676608000","2428678219968000","92521075046400","2296019325600","37297260000","390733200","2527200","9126","14"],["101421602465863680000","473300811507363840000","461468291219679744000","175797444274163712000","33572428594024320000","3662446755711744000","246510839326752000","10732444705382400","310727948731200","6057075024000","79318839600","684028800","3705156","11368","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","103","1810","40109","1069756","33299923","1183642118","47261918425","2093124561904","101778167892671","5388573485309818","308486434095022213","18983915986836677540","1249480584965869514539"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2","-1"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["4","-2","0"],["-8","-6","1"],["0","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","0"],["0","0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191085,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-dcd9ef61d075c3ec"]},{"id":191086,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-dcd9ef61d075c3ec"]},{"id":191087,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-dcd9ef61d075c3ec"]}],"region_exact":{},"incoming_edges":[{"id":"e-dcd9ef61d075c3ec","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-c05cf5b45147fec7","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":"7552b0469126429e70fe6b39747aeb7281833e1457ecbff3487ead01b49bf6e3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-66d7d2fd3a2f1f0d","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:            1;\n   2:            6,             2;\n   3:           60,            40,            3;\n   4:          840,           840,          126,            4;\n   5:        15120,         20160,         4536,          288,           5;\n   6:       332640,        554400,       166320,        15840,         550,         6;\n   7:      8648640,      17297280,      6486480,       823680,       42900,       936,        7;\n   8:    259459200,     605404800,    272432160,     43243200,     3003000,     98280,     1470,      8;\n   9:   8821612800,   23524300800,  12350257920,   2352430080,   204204000,   8910720,   199920,   2176,    9;\n  10: 335221286400, 1005663859200, 603398315520, 134088514560, 13967553600, 761866560, 22790880, 372096, 3078, 10;\n","recurrence_sentence":"The row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(8*n-4-(4*n^2-6*n-2)*x-n*x^2)*P(n-1,x) - (8*x-4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 4 - (4*n^2 - 6*n - 2)*x - n*x^2)*P[n - 1, x] - (8*x - 4*n*x^2)*D[P[n - 1, x], x])/(-2 + (n - 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n - 4 - (4*n^2 - 6*n - 2)*x - n*x^2)*v[i - 1] - (8*x - 4*n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 1)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-66d7d2fd3a2f1f0d","steps":["\u03b8+1"],"edges":["e-dcd9ef61d075c3ec"],"text":"A292219 \u2192 \u03b8+1"},"headline":"A292219 \u2192 \u03b8+1","chain":null}
