{"id":"n-893812c1efc3542a","revision":0,"first_index":0,"fingerprint":"f6-62e59b3741b0e3c5","name":"theta_plus_one of Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).","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"],["1","8"],["1","48","48"],["1","248","720","256"],["1","1248","7968","7168","1280"],["1","6248","79200","135680","57600","6144"],["1","31248","750288","2186240","1657600","405504","28672"],["1","156248","6940080","32422656","39110400","16472064","2609152","131072"],["1","781248","63398208","458508288","826990080","533643264","142098432","15728640","589824"],["1","3906248","575271360","6298731520","16351372800","15176060928","6042796032","1105723392","90243072","2621440"],["1","19531248","5200879728","84951623680","309466995200","397183868928","221891518464","59690188800","7953776640","498073600","11534336"],["1","97656248","46925105040","1132109133056","5685697036800","9826302824448","7400812683264","2745376702464","531080478720","53778841600","2664431616","50331648"],["1","488281248","422911882848","14967685956608","102317395290880","233643705090048","230876396929024","113448210923520","29879850958848","4350174822400","345868599296","13891534848","218103808"],["1","2441406248","3809136633120","196835447477760","1814234149728000","5397641304287232","6862247213645824","4345433074171904","1496552030797824","293755806023680","33321381789696","2134363865088","70883737600","939524096"],["1","12207031248","34296878135568","2579176212587520","31826157782764800","122058791308726272","196745173094486016","157387832127651840","68940665850101760","17520307181977600","2658702199881728","241448767193088","12722213224448","355140108288","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","97","1225","17665","284873","5059553","97841673","2041737985","45646726793","1086867990625","27424895224905","730266798373377","20446038173088073","600005644990402017"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","24","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","12","-16"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187062,"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-c6832d43a8409c1c"]},{"id":187063,"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-c6832d43a8409c1c"]},{"id":187064,"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-c6832d43a8409c1c"]},{"id":187065,"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-c6832d43a8409c1c"]}],"region_exact":{},"incoming_edges":[{"id":"e-c6832d43a8409c1c","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-893812c1efc3542a","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":"5ab7bef4c62a1ddb566d386bd6ed2e25540e4769b01a2a29655669fcf0d3cbd1","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-4c2a0f8cb422ddaa","role":"sequence","name":"Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).","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: 1,       8;\n  2: 1,      48,        48;\n  3: 1,     248,       720,        256;\n  4: 1,    1248,      7968,       7168,        1280;\n  5: 1,    6248,     79200,     135680,       57600,        6144;\n  6: 1,   31248,    750288,    2186240,     1657600,      405504,      28672;\n  7: 1,  156248,   6940080,   32422656,    39110400,    16472064,    2609152,     131072;\n  8: 1,  781248,  63398208,  458508288,   826990080,   533643264,  142098432,   15728640,   589824;\n  9: 1, 3906248, 575271360, 6298731520, 16351372800, 15176060928, 6042796032, 1105723392, 90243072, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+24*x-16*x^2)*P(n-2,x) + (12*x-16*x^2)*d/dx P(n-2,x) - (2-8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 24*x - 16*x^2)*P[n - 2, x] + (12*x - 16*x^2)*D[P[n - 2, x], x] - (2 - 8*x)*P[n - 1, x] + 4*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] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 24*x - 16*x^2)*v[i - 2] + (12*x - 16*x^2)*deriv(v[i - 2], x) - (2 - 8*x)*v[i - 1] + 4*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-4c2a0f8cb422ddaa","steps":["\u03b8+1"],"edges":["e-c6832d43a8409c1c"],"text":"A285061 \u2192 \u03b8+1"},"headline":"A285061 \u2192 \u03b8+1","chain":null}