{"id":"n-2057ae96e4f2c5de","revision":0,"first_index":0,"fingerprint":"f6-9776d3e5efdce8d6","name":"theta_plus_one of basis_map_A013609 of Exponential Riordan array (1, x(1+x/2)).","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","4"],["2","12","12"],["4","36","72","32"],["10","112","324","320","80"],["26","360","1380","2080","1200","192"],["76","1224","5760","11680","10800","4032","448"],["232","4312","24192","61600","78400","48384","12544","1024"],["764","15936","103152","315392","512400","446208","194432","36864","2304"],["2620","60912","447984","1599360","3160080","3568320","2239104","718848","103680","5120"],["9496","242320","1989360","8108160","18866400","26236224","21685440","10168320","2488320","281600","11264"],["35696","994224","9023520","41352960","110721600","183040704","188555136","118103040","42577920","8166400","743424","24576"],["140152","4220352","41885712","212960000","644608800","1235443968","1527586368","1206171648","587347200","166707200","25648128","1916928","53248"],["568504","18425056","198621072","1109643392","3746256800","8167775616","11798090304","11286832128","7000950528","2705331200","616918016","77635584","4845568","114688"],["2390480","82851552","963004224","5859817600","21821479680","53331358080","88189989888","99325264896","75312357120","37398000640","11669898240","2176032768","227741696","12042240","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","144","846","5238","34020","230688","1627452","11906028","90086904","703339200","5654689704","46732008456","396372474864"],"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":[["2","4","0","0"],["0","2","1","-4"],["0","0","1","0"]]},"terms":[{"coeff":{"coeffs":[["4","14","12","-4","-8"],["-2","3","14","2","-8"],["0","-3","1","6","0"],["0","0","-1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","4","2","-8","-8"],["0","2","5","-2","-8"],["0","0","1","2","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178315,"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-f594d72820bf7e11"]},{"id":178316,"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-f594d72820bf7e11"]},{"id":178317,"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-f594d72820bf7e11"]}],"region_exact":{},"incoming_edges":[{"id":"e-f594d72820bf7e11","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-2057ae96e4f2c5de","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":"62cfef745e555b40e279d954e189478b6c238599bafa1859b721c820c43b3f0b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f771467984d796d1","role":"sequence","name":"basis_map_A013609 of Exponential Riordan array (1, x(1+x/2)).","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,     4;\n  2:    2,    12,     12;\n  3:    4,    36,     72,      32;\n  4:   10,   112,    324,     320,      80;\n  5:   26,   360,   1380,    2080,    1200,     192;\n  6:   76,  1224,   5760,   11680,   10800,    4032,     448;\n  7:  232,  4312,  24192,   61600,   78400,   48384,   12544,   1024;\n  8:  764, 15936, 103152,  315392,  512400,  446208,  194432,  36864,   2304;\n  9: 2620, 60912, 447984, 1599360, 3160080, 3568320, 2239104, 718848, 103680, 5120;\n","recurrence_sentence":"The row polynomials satisfy (2+(2*n+4)*x+(n^2+n)*x^2-4*n*x^3)*P(n,x) = -(2*n-4+(3*n^2-3*n-14)*x+(n^3-n^2-14*n-12)*x^2-(6*n^2+2*n-4)*x^3+(8*n+8)*x^4)*P(n-1,x) + (1+(2*n+4)*x+(n^2+5*n+2)*x^2+(2*n^2-2*n-8)*x^3-(8*n+8)*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[(-(2*n - 4 + (3*n^2 - 3*n - 14)*x + (n^3 - n^2 - 14*n - 12)*x^2 - (6*n^2 + 2*n - 4)*x^3 + (8*n + 8)*x^4)*P[n - 1, x] + (1 + (2*n + 4)*x + (n^2 + 5*n + 2)*x^2 + (2*n^2 - 2*n - 8)*x^3 - (8*n + 8)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 4)*x + (n^2 + n)*x^2 - 4*n*x^3)]];\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] = (-(2*n - 4 + (3*n^2 - 3*n - 14)*x + (n^3 - n^2 - 14*n - 12)*x^2 - (6*n^2 + 2*n - 4)*x^3 + (8*n + 8)*x^4)*v[i - 1] + (1 + (2*n + 4)*x + (n^2 + 5*n + 2)*x^2 + (2*n^2 - 2*n - 8)*x^3 - (8*n + 8)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 4)*x + (n^2 + n)*x^2 - 4*n*x^3)); 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-f6268da8d9636534","steps":["basis A013609","\u03b8+1"],"edges":["e-8afcaf02015990bf","e-f594d72820bf7e11"],"text":"A122848 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A122848 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}
