{"id":"n-f8754aa1558d3cbf","revision":0,"first_index":0,"fingerprint":"f6-24a6b574ee77e562","name":"theta_plus_one of reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","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"],["3","2"],["5","10","3"],["7","28","21","4"],["9","60","81","36","5"],["11","110","231","176","55","6"],["13","182","546","624","325","78","7"],["15","280","1134","1800","1375","540","105","8"],["17","408","2142","4488","4675","2652","833","136","9"],["19","570","3762","10032","13585","10374","4655","1216","171","10"],["21","770","6237","20592","35035","34398","20580","7616","1701","210","11"],["23","1012","9867","39468","82225","100464","76636","37536","11799","2300","253","12"],["25","1300","15015","71500","178750","265200","249900","155040","64125","17500","3025","300","13"],["27","1638","22113","123552","364650","644436","732564","558144","290871","103950","25047","3888","351","14"],["29","2030","31668","205088","704990","1461252","1967070","1798464","1145529","513590","161414","34800","4901","406","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","18","60","191","589","1775","5257","15360","44394","127171","361595","1021693","2871245","8031246"],"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":[["2","0"],["-4","1"],["0","-3"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["-2","0","0"],["-4","-5","-1"],["0","-1","0"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-6","-1"],["0","0","0","0"],["0","0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173591,"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-f0dcfa317a8836f6"]},{"id":173592,"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-f0dcfa317a8836f6"]},{"id":173593,"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-f0dcfa317a8836f6"]}],"region_exact":{},"incoming_edges":[{"id":"e-f0dcfa317a8836f6","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-f8754aa1558d3cbf","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":"9dab89f8dbdc078496f2444eaab6300bdda3eb14633cc53361be2e8b58cfc38c","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-fe40873ed3cfd4a4","role":"sequence","name":"reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","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":[{"id":"l-143889220c0d58cf","kind":"fingerprint_related","node_a_id":"n-cb54800b91762264","node_b_id":"n-f8754aa1558d3cbf","fingerprint":"f6-24a6b574ee77e562","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-24a6b574ee77e562"},"other_id":"n-cb54800b91762264","other":{"id":"n-cb54800b91762264","name":"theta_plus_one of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","first_index":1,"fingerprint":"f6-24a6b574ee77e562","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["5","10","3"]],"card_rows":[["1"],["3","2"],["5","10","3"],["7","28","21","4"]],"row_sums_preview":["1","5","18","60","191","589","1775","5257"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-318700a428d7b97b","steps":["\u03b8+1"],"edges":["e-1de9fee97176e47c"],"text":"A111125 \u2192 \u03b8+1"},"headline":"A111125 \u2192 \u03b8+1","chain":null}}],"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,  10,    3;\n  3:  7,  28,   21,     4;\n  4:  9,  60,   81,    36,     5;\n  5: 11, 110,  231,   176,    55,     6;\n  6: 13, 182,  546,   624,   325,    78,    7;\n  7: 15, 280, 1134,  1800,  1375,   540,  105,    8;\n  8: 17, 408, 2142,  4488,  4675,  2652,  833,  136,   9;\n  9: 19, 570, 3762, 10032, 13585, 10374, 4655, 1216, 171, 10;\n","recurrence_sentence":"The row polynomials satisfy -(4*n-2-(2*n^3-3*n^2+n)*x)*P(n,x) = -(4*n+2-(2*n^3-n^2-5*n)*x-(n^3-n)*x^2)*P(n-1,x) - (8*x-(4*n^2-6)*x^2-(n^2-1)*x^3)*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[(-(4*n + 2 - (2*n^3 - n^2 - 5*n)*x - (n^3 - n)*x^2)*P[n - 1, x] - (8*x - (4*n^2 - 6)*x^2 - (n^2 - 1)*x^3)*D[P[n - 1, x], x])/(-4*n + 2 + (2*n^3 - 3*n^2 + n)*x)]];\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] = (-(4*n + 2 - (2*n^3 - n^2 - 5*n)*x - (n^3 - n)*x^2)*v[i - 1] - (8*x - (4*n^2 - 6)*x^2 - (n^2 - 1)*x^3)*deriv(v[i - 1], x))/(-4*n + 2 + (2*n^3 - 3*n^2 + 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-9496336e2b29e28e","steps":["reverse","\u03b8+1"],"edges":["e-41a825a32145fbe0","e-f0dcfa317a8836f6"],"text":"A082985 \u2192 reverse \u2192 \u03b8+1"},"headline":"A082985 \u2192 reverse \u2192 \u03b8+1","chain":null}