{"id":"n-c3dfc3a32f923c60","revision":0,"first_index":1,"fingerprint":"f6-b99dd02e1a9529cf","name":"theta_plus_one of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","20","3"],["1","118","177","4"],["1","616","3186","1232","5"],["1","3114","40686","54248","7785","6"],["1","15612","445581","1410448","742635","46836","7"],["1","78110","4502007","28435660","35544575","9004014","273385","8"],["1","390608","43447188","495719776","1208515550","743579664","101376772","1562432","9"],["1","1953106","408016140","7892596688","33557408470","40268890164","13812044204","1088043040","8788977","10"],["1","9765604","3768824007","118380381024","817119590490","1691293386888","1143967426686","236760762048","11306472021","48828020","11"],["1","48828102","34461407085","1704773209620","18182321850450","60030748839348","70035873645906","29091714960720","3835739721645","114871356950","268554561","12"],["1","244140600","313155592038","23862147807920","379421366354475","1893390526021968","3501793682295300","2524520701362624","682958459438055","59655369519800","1148237170806","1464843600","13"],["1","1220703098","2834879818842","327327093867704","7553787564142375","54786287154098538","151604554854459084","173262348405096096","82179430731147807","15107575128284750","900149508136186","11339519275368","7934570137","14"],["1","6103515596","25603640047281","4425345009410672","145189902151140205","1485900198083990748","5899385926794270813","10049216207495573568","7584924763021205331","2476500330139984580","319417784732508451","13276035028232016","110949106871551","42724609172","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","24","300","5040","105840","2661120","77837760","2594592000","97037740800","4022655436800","183030822374400","9067065354547200","485735643993600000","27978373094031360000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["-7"],["4"]]},"terms":[{"coeff":{"coeffs":[["-9","198","231"],["12","-336","-524"],["0","96","336"],["0","0","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-36","-48","84"],["0","48","112","-160"],["0","0","-64","64"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","98"],["-8","-112"],["0","32"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-28","28"],["0","16","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187539,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-d9483910992974a7"]},{"id":187540,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-d9483910992974a7"]},{"id":187541,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-d9483910992974a7"]},{"id":187542,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-d9483910992974a7"]}],"region_exact":{},"incoming_edges":[{"id":"e-d9483910992974a7","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-c3dfc3a32f923c60","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":"bf8455f81f1bc2753126e30cdc62791092ffdf3a33a5cb799a5a4c60a5ebaa42","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d023ddeb9da11712","role":"sequence","name":"Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-aa2e97bb4f1e6351","kind":"fingerprint_related","node_a_id":"n-c3dfc3a32f923c60","node_b_id":"n-85a3dd1215d7617f","fingerprint":"f6-b99dd02e1a9529cf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b99dd02e1a9529cf"},"other_id":"n-85a3dd1215d7617f","other":{"id":"n-85a3dd1215d7617f","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.","first_index":0,"fingerprint":"f6-b99dd02e1a9529cf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["4","80","12"]],"card_rows":[["1"],["2","4"],["4","80","12"],["8","944","1416","32"]],"row_sums_preview":["1","6","96","2400","80640","3386880","170311680","9963233280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fb16da45c5e69d3d","steps":["\u03b8+1"],"edges":["e-9986c9ab5bbfd3e4"],"text":"A257618 \u2192 \u03b8+1"},"headline":"A257618 \u2192 \u03b8+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,       2;\n   3: 1,      20,         3;\n   4: 1,     118,       177,          4;\n   5: 1,     616,      3186,       1232,           5;\n   6: 1,    3114,     40686,      54248,        7785,           6;\n   7: 1,   15612,    445581,    1410448,      742635,       46836,           7;\n   8: 1,   78110,   4502007,   28435660,    35544575,     9004014,      273385,          8;\n   9: 1,  390608,  43447188,  495719776,  1208515550,   743579664,   101376772,    1562432,       9;\n  10: 1, 1953106, 408016140, 7892596688, 33557408470, 40268890164, 13812044204, 1088043040, 8788977, 10;\n","recurrence_sentence":"The row polynomials satisfy (4*n-7)*P(n,x) = (12*n-9+(96*n^2-336*n+198)*x-(64*n^3-336*n^2+524*n-231)*x^2)*P(n-2,x) + ((48*n-36)*x-(64*n^2-112*n+48)*x^2+(64*n^2-160*n+84)*x^3)*d/dx P(n-2,x) - (8*n-2-(32*n^2-112*n+98)*x)*P(n-1,x) + ((16*n-28)*x-(16*n-28)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n - 9 + (96*n^2 - 336*n + 198)*x - (64*n^3 - 336*n^2 + 524*n - 231)*x^2)*P[n - 2, x] + ((48*n - 36)*x - (64*n^2 - 112*n + 48)*x^2 + (64*n^2 - 160*n + 84)*x^3)*D[P[n - 2, x], x] - (8*n - 2 - (32*n^2 - 112*n + 98)*x)*P[n - 1, x] + ((16*n - 28)*x - (16*n - 28)*x^2)*D[P[n - 1, x], x])/(4*n - 7)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = ((12*n - 9 + (96*n^2 - 336*n + 198)*x - (64*n^3 - 336*n^2 + 524*n - 231)*x^2)*v[i - 2] + ((48*n - 36)*x - (64*n^2 - 112*n + 48)*x^2 + (64*n^2 - 160*n + 84)*x^3)*deriv(v[i - 2], x) - (8*n - 2 - (32*n^2 - 112*n + 98)*x)*v[i - 1] + ((16*n - 28)*x - (16*n - 28)*x^2)*deriv(v[i - 1], x))/(4*n - 7)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d023ddeb9da11712","steps":["\u03b8+1"],"edges":["e-d9483910992974a7"],"text":"A142459 \u2192 \u03b8+1"},"headline":"A142459 \u2192 \u03b8+1","chain":null}