{"id":"n-c443535979497fee","revision":0,"first_index":0,"fingerprint":"f6-ef846d88422ec39f","name":"theta_plus_one_squared of shift_x_plus_one of chebyshev_t","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"],["1","4"],["1","16","18"],["1","36","108","64"],["1","64","360","512","200"],["1","100","900","2240","2000","576"],["1","144","1890","7168","10800","6912","1568"],["1","196","3528","18816","42000","44352","21952","4096"],["1","256","6048","43008","132000","202752","163072","65536","10368"],["1","324","9720","88704","356400","741312","856128","552960","186624","25600"],["1","400","14850","168960","858000","2306304","3567200","3276800","1762560","512000","61952"],["1","484","21780","302016","1887600","6342336","12556544","15319040","11632896","5350400","1362944","147456"],["1","576","30888","512512","3861000","15814656","38811136","60162048","60279552","38912000","15611904","3538944","346112"],["1","676","42588","832832","7436000","36406656","108116736","206389248","261211392","221312000","124027904","44089344","8998912","802816"],["1","784","57330","1304576","13613600","78414336","276529344","635043840","984566016","1048678400","768018944","379846656","121139200","22478848","1843200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","35","209","1137","5817","28483","134941","623041","2817773","12529027","54923497","237881329","1019667105","4331535075"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"denominator":{"coeffs":[["-9","6","0"],["9","-28","0"],["0","33","-4"],["0","-11","8"],["0","0","-5"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-9","-30","-20","-4"],["9","8","1","0"],["0","33","34","7"],["0","-11","-11","-1"],["0","0","-5","-3"],["0","0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","18","35","21","4"],["0","0","22","19","4"],["0","0","-22","-17","-3"],["0","0","0","-4","-2"],["0","0","0","2","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174567,"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-308d9c02fc1c4e65"]},{"id":174568,"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-308d9c02fc1c4e65"]},{"id":174569,"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-308d9c02fc1c4e65"]}],"region_exact":{},"incoming_edges":[{"id":"e-308d9c02fc1c4e65","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-c443535979497fee","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":"b9380de68187919d03dd68118e88d0304e7acec6288abd876b48ad1144e3ccab","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-5fc5274bb6446a33","role":"sequence","name":"shift_x_plus_one of chebyshev_t","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,   4;\n  2: 1,  16,   18;\n  3: 1,  36,  108,    64;\n  4: 1,  64,  360,   512,    200;\n  5: 1, 100,  900,  2240,   2000,    576;\n  6: 1, 144, 1890,  7168,  10800,   6912,   1568;\n  7: 1, 196, 3528, 18816,  42000,  44352,  21952,   4096;\n  8: 1, 256, 6048, 43008, 132000, 202752, 163072,  65536,  10368;\n  9: 1, 324, 9720, 88704, 356400, 741312, 856128, 552960, 186624, 25600;\n","recurrence_sentence":"The row polynomials satisfy (9*n-9-(11*n^3-33*n^2+28*n-6)*x+(n^5-5*n^4+8*n^3-4*n^2)*x^2)*P(n,x) = (9*n-9-(11*n^3-33*n^2-8*n+30)*x+(n^5-5*n^4-11*n^3+34*n^2+n-20)*x^2+(n^5-3*n^4-n^3+7*n^2-4)*x^3)*P(n-1,x) + (18*x-(22*n^2-22*n-35)*x^2+(2*n^4-4*n^3-17*n^2+19*n+21)*x^3+(n^4-2*n^3-3*n^2+4*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n - 9 - (11*n^3 - 33*n^2 - 8*n + 30)*x + (n^5 - 5*n^4 - 11*n^3 + 34*n^2 + n - 20)*x^2 + (n^5 - 3*n^4 - n^3 + 7*n^2 - 4)*x^3)*P[n - 1, x] + (18*x - (22*n^2 - 22*n - 35)*x^2 + (2*n^4 - 4*n^3 - 17*n^2 + 19*n + 21)*x^3 + (n^4 - 2*n^3 - 3*n^2 + 4*n + 4)*x^4)*D[P[n - 1, x], x])/(9*n - 9 - (11*n^3 - 33*n^2 + 28*n - 6)*x + (n^5 - 5*n^4 + 8*n^3 - 4*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; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n - 9 - (11*n^3 - 33*n^2 - 8*n + 30)*x + (n^5 - 5*n^4 - 11*n^3 + 34*n^2 + n - 20)*x^2 + (n^5 - 3*n^4 - n^3 + 7*n^2 - 4)*x^3)*v[i - 1] + (18*x - (22*n^2 - 22*n - 35)*x^2 + (2*n^4 - 4*n^3 - 17*n^2 + 19*n + 21)*x^3 + (n^4 - 2*n^3 - 3*n^2 + 4*n + 4)*x^4)*deriv(v[i - 1], x))/(9*n - 9 - (11*n^3 - 33*n^2 + 28*n - 6)*x + (n^5 - 5*n^4 + 8*n^3 - 4*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-9ae8d431f7b33f22","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-298ee5f2391f30e9","e-308d9c02fc1c4e65"],"text":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}