{"id":"n-c5b24384f32713ef","revision":0,"first_index":0,"fingerprint":"f6-3444b903bbab1a5c","name":"theta_plus_one of Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].","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","6"],["1","30","27"],["1","126","324","108"],["1","510","2835","2376","405"],["1","2046","22140","35100","14175","1458"],["1","8190","164187","439560","315900","74358","5103"],["1","32766","1186164","5052348","5755050","2326968","357210","17496"],["1","131070","8450595","55268136","93761955","57949668","14931378","1609632","59049"],["1","524286","59743980","586483740","1426160925","1264737726","486520020","86605200","6908733","196830"],["1","2097150","420567147","6103813320","20739406050","25369982946","13670462421","3573383040","465010875","28540350","649539"],["1","8388606","2953407204","62720401788","292501578600","480581588916","348533726310","125484583752","23685439635","2349166050","114318864","2125764"],["1","33554430","20711599155","639017646696","4037722028505","8742311105616","8304176361096","3955633618464","1015646461038","144728114850","11296132749","446410440","6908733"],["1","134217726","145132189020","6473022863580","54886702545675","154412776992474","188377439730480","115495401415680","38741424988266","7437875419260","827782894224","52147116684","1706457051","22320522"],["1","536870910","1016529302907","65310757391880","737800968832200","2668196561044014","4119616074352779","3186764338792320","1358332604484570","337398595033500","50206258604502","4482109621080","232617040110","6405989814","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","58","559","6127","74920","1007299","14728003","232161484","3917881441","70373912839","1338934725490","26871729941773","566851569150643","12529364429105122"],"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","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","12","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","6","-9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185046,"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-bd39dc2008c7dad7"]},{"id":185047,"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-bd39dc2008c7dad7"]},{"id":185048,"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-bd39dc2008c7dad7"]},{"id":185049,"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-bd39dc2008c7dad7"]}],"region_exact":{},"incoming_edges":[{"id":"e-bd39dc2008c7dad7","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-c5b24384f32713ef","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":"af4ff3798f314471cb16598498e875dfc5ce61e7b983c7a929272378f18d6fd4","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-315041bb871f134e","role":"sequence","name":"Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].","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,      6;\n  2: 1,     30,       27;\n  3: 1,    126,      324,       108;\n  4: 1,    510,     2835,      2376,        405;\n  5: 1,   2046,    22140,     35100,      14175,       1458;\n  6: 1,   8190,   164187,    439560,     315900,      74358,      5103;\n  7: 1,  32766,  1186164,   5052348,    5755050,    2326968,    357210,    17496;\n  8: 1, 131070,  8450595,  55268136,   93761955,   57949668,  14931378,  1609632,   59049;\n  9: 1, 524286, 59743980, 586483740, 1426160925, 1264737726, 486520020, 86605200, 6908733, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+12*x-9*x^2)*P(n-2,x) + (6*x-9*x^2)*d/dx P(n-2,x) - (1-6*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 12*x - 9*x^2)*P[n - 2, x] + (6*x - 9*x^2)*D[P[n - 2, x], x] - (1 - 6*x)*P[n - 1, x] + 3*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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 12*x - 9*x^2)*v[i - 2] + (6*x - 9*x^2)*deriv(v[i - 2], x) - (1 - 6*x)*v[i - 1] + 3*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-315041bb871f134e","steps":["\u03b8+1"],"edges":["e-bd39dc2008c7dad7"],"text":"A282629 \u2192 \u03b8+1"},"headline":"A282629 \u2192 \u03b8+1","chain":null}