{"id":"n-812ee6d44fe8342a","revision":0,"first_index":0,"fingerprint":"f6-ed4651651bd37d3d","name":"theta_plus_one of Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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"],["2","2"],["3","10","3"],["4","26","27","4"],["5","52","105","56","5"],["6","90","288","300","100","6"],["7","142","648","1068","700","162","7"],["8","210","1281","3000","3115","1428","245","8"],["9","296","2310","7200","10615","7728","2646","352","9"],["10","402","3888","15432","30230","31266","17052","4560","486","10"],["11","530","6201","30360","75575","103818","80535","34416","7425","650","11"],["12","682","9471","55836","171050","298326","307461","187176","64746","11550","847","12"],["13","860","13959","97240","357830","767124","1003555","812240","400950","115060","17303","1080","13"],["14","1066","19968","161876","702130","1805700","2902088","2981992","1958814","803660","195052","25116","1352","14"],["15","1302","27846","259428","1306305","3954132","7617981","9615336","8022105","4384380","1524237","317772","35490","1666","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","16","61","223","790","2734","9295","31165","103336","339532","1107169","3587227","11558842","37068010"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"],["3","10","3"]],"recurrence":{"denominator":{"coeffs":[["4"]]},"terms":[{"coeff":{"coeffs":[["8","6","-6","4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1","-3","-3","-4"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-20","-8","-12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","6","8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["14","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174354,"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-7f1e30d9a4205233"]},{"id":174355,"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-7f1e30d9a4205233"]},{"id":174356,"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-7f1e30d9a4205233"]}],"region_exact":{},"incoming_edges":[{"id":"e-7f1e30d9a4205233","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-812ee6d44fe8342a","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":"91f70239de572f49eb1e6ec6d15c5d5cce6cfdd38e1b9296b052b6ac195fdf05","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-9fa98aec39d32ffa","role":"sequence","name":"Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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:  2,   2;\n  2:  3,  10,    3;\n  3:  4,  26,   27,     4;\n  4:  5,  52,  105,    56,     5;\n  5:  6,  90,  288,   300,   100,     6;\n  6:  7, 142,  648,  1068,   700,   162,     7;\n  7:  8, 210, 1281,  3000,  3115,  1428,   245,    8;\n  8:  9, 296, 2310,  7200, 10615,  7728,  2646,  352,   9;\n  9: 10, 402, 3888, 15432, 30230, 31266, 17052, 4560, 486, 10;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = (8+6*x-6*x^2+4*x^3)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (20+8*x+12*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (14+12*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 10*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 3 + 10*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + 6*x - 6*x^2 + 4*x^3)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (20 + 8*x + 12*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (14 + 12*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 10*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((8 + 6*x - 6*x^2 + 4*x^3)*v[i - 3] + (1 - 3*x - 3*x^2 - 4*x^3)*deriv(v[i - 3], x) - (20 + 8*x + 12*x^2)*v[i - 2] - (2 - 6*x - 8*x^2)*deriv(v[i - 2], x) + (14 + 12*x)*v[i - 1] + (1 - 4*x)*deriv(v[i - 1], x))/(4)); 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-9fa98aec39d32ffa","steps":["\u03b8+1"],"edges":["e-7f1e30d9a4205233"],"text":"A104029 \u2192 \u03b8+1"},"headline":"A104029 \u2192 \u03b8+1","chain":null}