{"id":"n-a82b0b9b0c966913","revision":0,"first_index":0,"fingerprint":"f6-b0bcc67e2c05872b","name":"theta_plus_one of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+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"],["3","8"],["5","80","48"],["7","280","1008","256"],["9","672","6048","9216","1280"],["11","1320","22176","84480","70400","6144"],["13","2288","61776","439296","915200","479232","28672"],["15","3640","144144","1647360","6406400","8386560","3010560","131072"],["17","5440","297024","4978688","31116800","76038144","68239360","17825792","589824"],["19","7752","558144","12899328","118243840","464375808","777928704","508035072","100859904","2621440"],["21","10640","976752","29767680","376230400","2167087104","5834465280","7112491008","3530096640","550502400","11534336"],["23","14168","1615152","62760192","1046003200","8307167232","32802660352","64266436608","59540963328","23212851200","2918187008","50331648"],["25","18400","2550240","123059200","2615008000","27386265600","149103001600","428442910720","637938892800","464257024000","145909350400","15099494400","218103808"],["27","23400","3875040","227335680","5999136000","80104826880","575111577600","2278537297920","4975923363840","5819793408000","3414278799360","883320422400","76554436608","939524096"],["29","29232","5700240","399559680","12819206400","212565749760","1945794170880","10165781790720","30609467965440","52507469414400","49507042590720","23908539432960","5180183543808","381446782976","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","133","1551","17225","184531","1926477","19729751","199091089","1985530011","19613162261","192208990111","1871095679193","18110794026851","174435542469085"],"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":[["1","0"],["-2","4"],["0","-16"],["0","16"]]},"terms":[{"coeff":{"coeffs":[["-1","40","48"],["-2","-52","-208"],["0","0","0"],["0","16","192"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","8","32"],["0","0","32","-128"],["0","0","32","-128"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184896,"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-22702f4bae078683"]},{"id":184897,"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-22702f4bae078683"]},{"id":184898,"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-22702f4bae078683"]},{"id":184899,"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-22702f4bae078683"]}],"region_exact":{},"incoming_edges":[{"id":"e-22702f4bae078683","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-a82b0b9b0c966913","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":"d613961fcd2c82bfa2990edc482fa23e005b49a5e5af5dafa34858866eee218f","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-9aec4822212db1dc","role":"sequence","name":"Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+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:  3,    8;\n  2:  5,   80,     48;\n  3:  7,  280,   1008,      256;\n  4:  9,  672,   6048,     9216,      1280;\n  5: 11, 1320,  22176,    84480,     70400,      6144;\n  6: 13, 2288,  61776,   439296,    915200,    479232,     28672;\n  7: 15, 3640, 144144,  1647360,   6406400,   8386560,   3010560,    131072;\n  8: 17, 5440, 297024,  4978688,  31116800,  76038144,  68239360,  17825792,    589824;\n  9: 19, 7752, 558144, 12899328, 118243840, 464375808, 777928704, 508035072, 100859904, 2621440;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-1-(16*n^3-16*n^2+4*n)*x)*P(n,x) = -(2*n+1-(16*n^3-52*n+40)*x-(192*n^3-208*n+48)*x^2)*P(n-1,x) - (4*x-(32*n^2+32*n+8)*x^2+(128*n^2+128*n-32)*x^3)*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 + 1 - (16*n^3 - 52*n + 40)*x - (192*n^3 - 208*n + 48)*x^2)*P[n - 1, x] - (4*x - (32*n^2 + 32*n + 8)*x^2 + (128*n^2 + 128*n - 32)*x^3)*D[P[n - 1, x], x])/(-2*n + 1 + (16*n^3 - 16*n^2 + 4*n)*x)]];\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 + 1 - (16*n^3 - 52*n + 40)*x - (192*n^3 - 208*n + 48)*x^2)*v[i - 1] - (4*x - (32*n^2 + 32*n + 8)*x^2 + (128*n^2 + 128*n - 32)*x^3)*deriv(v[i - 1], x))/(-2*n + 1 + (16*n^3 - 16*n^2 + 4*n)*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-9aec4822212db1dc","steps":["\u03b8+1"],"edges":["e-22702f4bae078683"],"text":"A085841 \u2192 \u03b8+1"},"headline":"A085841 \u2192 \u03b8+1","chain":null}
