{"id":"n-ab0b2de667036ea3","revision":0,"first_index":0,"fingerprint":"f6-57cbed3505751c46","name":"theta_plus_one of Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","2"],["4","6"],["8","16","3"],["16","40","15"],["32","96","54","4"],["64","224","168","28"],["128","512","480","128","5"],["256","1152","1296","480","45"],["512","2560","3360","1600","250","6"],["1024","5632","8448","4928","1100","66"],["2048","12288","20736","14336","4200","432","7"],["4096","26624","49920","39936","14560","2184","91"],["8192","57344","118272","107520","47040","9408","686","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","10","27","71","186","484","1253","3229","8288","21198","54047","137411","348470"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["-4","-1"],["4","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2"],["-8","-10","-1"],["8","8","2"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","-4"],["0","-8","-14","-6"],["0","0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172619,"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-9264fe92ff22841c"]},{"id":172620,"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-9264fe92ff22841c"]},{"id":172621,"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-9264fe92ff22841c"]}],"region_exact":{},"incoming_edges":[{"id":"e-9264fe92ff22841c","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-ab0b2de667036ea3","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":"4a47fcb771ab91775912589e8f94762955604cf12e3326ecc18a94f25ee85601","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-495335f6a4ca3289","role":"sequence","name":"Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).","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;\n  2:   2,    2;\n  3:   4,    6;\n  4:   8,   16,    3;\n  5:  16,   40,   15;\n  6:  32,   96,   54,   4;\n  7:  64,  224,  168,  28;\n  8: 128,  512,  480, 128,  5;\n  9: 256, 1152, 1296, 480, 45;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-4*n+(n^3-n)*x)*P(n,x) = (8*n^2-8*n+(2*n^3+8*n^2-10*n)*x+(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (8*n*x+(2*n^2+14*n+4)*x^2+(2*n^2+6*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n + (2*n^3 + 8*n^2 - 10*n)*x + (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (8*n*x + (2*n^2 + 14*n + 4)*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 4*n + (n^3 - n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n + (2*n^3 + 8*n^2 - 10*n)*x + (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (8*n*x + (2*n^2 + 14*n + 4)*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 4*n + (n^3 - 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-495335f6a4ca3289","steps":["\u03b8+1"],"edges":["e-9264fe92ff22841c"],"text":"A201701 \u2192 \u03b8+1"},"headline":"A201701 \u2192 \u03b8+1","chain":null}