{"id":"n-ec55c0929287c3cd","revision":1,"first_index":0,"fingerprint":"f6-9913b280f2bd95b5","name":"theta_plus_one of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","2"],["5","4"],["11","14","3"],["21","32","9"],["43","82","36","4"],["85","188","102","16"],["171","438","297","72","5"],["341","984","783","240","25"],["683","2202","2034","780","125","6"],["1365","4852","5076","2304","475","36"],["2731","10622","12447","6576","1700","198","7"],["5461","23056","29877","17952","5530","840","49"],["10923","49762","70704","47700","17150","3276","294","8"],["21845","106796","165042","123440","50650","11592","1372","64"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","5","9","28","62","165","391","983","2373","5830","14108","34281","82765","199817","480801"],"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":[["36","22","2"],["36","31","5"],["0","9","4"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["36","22","2","0"],["180","245","101","12"],["0","63","52","10"],["0","0","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-324","-702","-554","-190","-24"],["0","-162","-243","-121","-20"],["0","0","-18","-17","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163312,"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-42735f4be786e5b0"]},{"id":163313,"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-42735f4be786e5b0"]},{"id":163314,"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-42735f4be786e5b0"]}],"region_exact":{},"incoming_edges":[{"id":"e-42735f4be786e5b0","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-ec55c0929287c3cd","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":"81e3aff60622d7c1647b929c41e5d8f45653e5e31beb7ef2e5775de9f951c2c8","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-ac04e52510a48bb5","role":"sequence","name":"Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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,    2;\n  2:   5,    4;\n  3:  11,   14,    3;\n  4:  21,   32,    9;\n  5:  43,   82,   36,   4;\n  6:  85,  188,  102,  16;\n  7: 171,  438,  297,  72,   5;\n  8: 341,  984,  783, 240,  25;\n  9: 683, 2202, 2034, 780, 125, 6;\n","recurrence_sentence":"The row polynomials satisfy (36*n+36+(9*n^2+31*n+22)*x+(n^3+4*n^2+5*n+2)*x^2)*P(n,x) = (180*n+36+(63*n^2+245*n+22)*x+(5*n^3+52*n^2+101*n+2)*x^2+(2*n^3+10*n^2+12*n)*x^3)*P(n-1,x) - (324+(162*n+702)*x+(18*n^2+243*n+554)*x^2+(17*n^2+121*n+190)*x^3+(4*n^2+20*n+24)*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[((180*n + 36 + (63*n^2 + 245*n + 22)*x + (5*n^3 + 52*n^2 + 101*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*P[n - 1, x] - (324 + (162*n + 702)*x + (18*n^2 + 243*n + 554)*x^2 + (17*n^2 + 121*n + 190)*x^3 + (4*n^2 + 20*n + 24)*x^4)*D[P[n - 1, x], x])/(36*n + 36 + (9*n^2 + 31*n + 22)*x + (n^3 + 4*n^2 + 5*n + 2)*x^2)]];\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] = ((180*n + 36 + (63*n^2 + 245*n + 22)*x + (5*n^3 + 52*n^2 + 101*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*v[i - 1] - (324 + (162*n + 702)*x + (18*n^2 + 243*n + 554)*x^2 + (17*n^2 + 121*n + 190)*x^3 + (4*n^2 + 20*n + 24)*x^4)*deriv(v[i - 1], x))/(36*n + 36 + (9*n^2 + 31*n + 22)*x + (n^3 + 4*n^2 + 5*n + 2)*x^2)); 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-75ccd0453a9b6cec","steps":["x \u21a6 x+2 (shift +1)","\u03b8+1"],"edges":["e-7fc1b55fb0aa373a","e-42735f4be786e5b0"],"text":"A011973 \u2192 x \u21a6 x+2 (shift +1) \u2192 \u03b8+1"},"headline":"A011973 \u2192 x \u21a6 x+2 (shift +1) \u2192 \u03b8+1","chain":null}