{"id":"n-34c3fca0a85f38f6","revision":0,"first_index":1,"fingerprint":"f6-ee33b1416b29d17c","name":"theta_plus_one_squared of reversal of Riordan array (1, 3+x).","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"],["9","4"],["27","24"],["81","108","9"],["243","432","81"],["729","1620","486","16"],["2187","5832","2430","192"],["6561","20412","10935","1440","25"],["19683","69984","45927","8640","375"],["59049","236196","183708","45360","3375","36"],["177147","787320","708588","217728","23625","648"],["531441","2598156","2657205","979776","141750","6804","49"],["1594323","8503056","9743085","4199040","765450","54432","1029"],["4782969","27634932","35075106","17321040","3827250","367416","12348","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","13","51","198","756","2851","10641","39373","144609","527724","1915056","6915181","24860415","89021125"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","-27","0"],["-243","189","-3"],["243","-243","9"],["0","81","-9"],["0","0","3"]]},"terms":[{"coeff":{"coeffs":[["0","-81","72","0"],["-729","-729","-9","-4"],["729","-81","-171","-6"],["0","243","63","0"],["0","0","9","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","81","36","0"],["0","-729","-324","-9","-4"],["0","0","-243","-126","-8"],["0","0","0","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174537,"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-94582978931fea25"]},{"id":174538,"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-94582978931fea25"]},{"id":174539,"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-94582978931fea25"]}],"region_exact":{},"incoming_edges":[{"id":"e-94582978931fea25","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-34c3fca0a85f38f6","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. 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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":"ac977246df3318e40a78ad1cbf054cf0cdc8861f3d01d6c3903cc68a413f9c79","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4598f43d5daedb7a","role":"sequence","name":"reversal of Riordan array (1, 3+x).","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":[],"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:     3;\n   3:     9,     4;\n   4:    27,    24;\n   5:    81,   108,     9;\n   6:   243,   432,    81;\n   7:   729,  1620,   486,   16;\n   8:  2187,  5832,  2430,  192;\n   9:  6561, 20412, 10935, 1440,  25;\n  10: 19683, 69984, 45927, 8640, 375;\n","recurrence_sentence":"The row polynomials satisfy (243*n^2-243*n+(81*n^3-243*n^2+189*n-27)*x+(3*n^4-9*n^3+9*n^2-3*n)*x^2)*P(n,x) = (729*n^2-729*n+(243*n^3-81*n^2-729*n-81)*x+(9*n^4+63*n^3-171*n^2-9*n+72)*x^2+(2*n^4-6*n^2-4*n)*x^3)*P(n-1,x) - (729*n*x+(243*n^2+324*n-81)*x^2+(9*n^3+126*n^2+9*n-36)*x^3+(4*n^3+8*n^2+4*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((729*n^2 - 729*n + (243*n^3 - 81*n^2 - 729*n - 81)*x + (9*n^4 + 63*n^3 - 171*n^2 - 9*n + 72)*x^2 + (2*n^4 - 6*n^2 - 4*n)*x^3)*P[n - 1, x] - (729*n*x + (243*n^2 + 324*n - 81)*x^2 + (9*n^3 + 126*n^2 + 9*n - 36)*x^3 + (4*n^3 + 8*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(243*n^2 - 243*n + (81*n^3 - 243*n^2 + 189*n - 27)*x + (3*n^4 - 9*n^3 + 9*n^2 - 3*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((729*n^2 - 729*n + (243*n^3 - 81*n^2 - 729*n - 81)*x + (9*n^4 + 63*n^3 - 171*n^2 - 9*n + 72)*x^2 + (2*n^4 - 6*n^2 - 4*n)*x^3)*v[i - 1] - (729*n*x + (243*n^2 + 324*n - 81)*x^2 + (9*n^3 + 126*n^2 + 9*n - 36)*x^3 + (4*n^3 + 8*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(243*n^2 - 243*n + (81*n^3 - 243*n^2 + 189*n - 27)*x + (3*n^4 - 9*n^3 + 9*n^2 - 3*n)*x^2)); 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-5d39164c4108e043","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-035822f9cf512625","e-94582978931fea25"],"text":"A099097 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A099097 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}