{"id":"n-20111fc1e9216d23","revision":0,"first_index":0,"fingerprint":"f6-8ae3b92bdc747eb3","name":"theta_plus_one of basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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"],["4","8"],["15","64","48"],["56","368","576","256"],["209","1856","4464","4096","1280"],["780","8728","28416","39936","25600","6144"],["2911","39296","161568","307200","300800","147456","28672"],["10864","171744","853632","2050560","2713600","2027520","802816","131072"],["40545","734592","4283424","12447744","20806400","20987904","12644352","4194304","589824"],["151316","3090984","20687616","70585344","142750720","181794816","147718144","74448896","21233664","2621440"],["564719","12839872","97012944","380227584","903123200","1391394816","1426604032","968884224","419364864","104857600","11534336"],["2107560","52786256","444403392","1967727360","5370880000","9718775808","12051873792","10322706432","6016204800","2280652800","507510784","50331648"],["7865521","215165632","1997318160","9860833280","30419033600","63263932416","92135178240","95416221696","70097633280","35756441600","12053381120","2415919104","218103808"],["29354524","870800440","8835863040","48127969280","165609420800","389348315136","651720523776","792531566592","703747325952","452290150400","205034356736","62209916928","11341398016","939524096"],["109552575","3502872320","38570936640","229775646720","872658496000","2289054547968","4331705720832","6054003867648","6311283720192","4901170053120","2798160707584","1141119123456","314723794944","52613349376","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","127","1256","11905","109604","987903","8761808","76729089","665082940","5716408191","48785960632","413857027457","3492636485716","29342478921215"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-12","-8","0"],["0","0","0"],["0","0","8"]]},"terms":[{"coeff":{"coeffs":[["-24","-52","-32","0"],["-24","-64","-32","16"],["0","-12","16","32"],["0","0","16","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-9","-36","-44","-16","0"],["0","0","12","32","16"],["0","0","12","32","16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183029,"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-291a46f7e61257ba"]},{"id":183030,"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-291a46f7e61257ba"]},{"id":183031,"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-291a46f7e61257ba"]},{"id":183032,"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-291a46f7e61257ba"]}],"region_exact":{},"incoming_edges":[{"id":"e-291a46f7e61257ba","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-20111fc1e9216d23","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":"aa01ee226cc018bb255cb2dc451c2d5fe66399836f4c918bf4e5e1fbfc381d75","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-1f449c58b1930aa5","role":"sequence","name":"basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:      4,       8;\n  2:     15,      64,       48;\n  3:     56,     368,      576,      256;\n  4:    209,    1856,     4464,     4096,      1280;\n  5:    780,    8728,    28416,    39936,     25600,      6144;\n  6:   2911,   39296,   161568,   307200,    300800,    147456,     28672;\n  7:  10864,  171744,   853632,  2050560,   2713600,   2027520,    802816,   131072;\n  8:  40545,  734592,  4283424, 12447744,  20806400,  20987904,  12644352,  4194304,   589824;\n  9: 151316, 3090984, 20687616, 70585344, 142750720, 181794816, 147718144, 74448896, 21233664, 2621440;\n","recurrence_sentence":"The row polynomials satisfy -(12*n+8*n*x-8*n^3*x^2)*P(n,x) = -(24*n+24+(12*n^2+64*n+52)*x-(16*n^3+16*n^2-32*n-32)*x^2-(16*n^3+32*n^2+16*n)*x^3)*P(n-1,x) - (9+36*x-(12*n^2+12*n-44)*x^2-(32*n^2+32*n-16)*x^3-(16*n^2+16*n)*x^4)*d/dx 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[(-(24*n + 24 + (12*n^2 + 64*n + 52)*x - (16*n^3 + 16*n^2 - 32*n - 32)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*P[n - 1, x] - (9 + 36*x - (12*n^2 + 12*n - 44)*x^2 - (32*n^2 + 32*n - 16)*x^3 - (16*n^2 + 16*n)*x^4)*D[P[n - 1, x], x])/(-12*n - 8*n*x + 8*n^3*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] = (-(24*n + 24 + (12*n^2 + 64*n + 52)*x - (16*n^3 + 16*n^2 - 32*n - 32)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*v[i - 1] - (9 + 36*x - (12*n^2 + 12*n - 44)*x^2 - (32*n^2 + 32*n - 16)*x^3 - (16*n^2 + 16*n)*x^4)*deriv(v[i - 1], x))/(-12*n - 8*n*x + 8*n^3*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-f287bed89e07609f","steps":["basis A013611","\u03b8+1"],"edges":["e-362ae2b737f320ab","e-291a46f7e61257ba"],"text":"A125662 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A125662 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}