{"id":"n-a59b31e2ece04346","revision":0,"first_index":0,"fingerprint":"f6-2ae4e9c0c8806563","name":"theta_plus_one of basis_map_A013609 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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"],["4","4"],["13","32","12"],["43","168","144","32"],["142","760","1044","512","80"],["469","3188","6060","4736","1600","192"],["1549","12768","31068","33664","18000","4608","448"],["5116","49564","147132","204736","151600","61056","12544","1024"],["16897","188120","659460","1123072","1068800","594816","191296","32768","2304"],["55807","701976","2838564","5724864","6677520","4774464","2116800","565248","82944","5120"],["184318","2584864","11845440","27625728","38221200","33509376","19070016","7004160","1596672","204800","11264"],["608761","9417068","48235404","127759392","204717600","213059520","148655808","70004736","21904128","4352000","495616","24576"],["2010601","34008384","192557448","571119488","1040836800","1255741056","1040364864","600391680","240579072","65484800","11523072","1179648","53248"],["6640564","121919668","756184104","2483232128","5074425200","6969613248","6696951744","4588941312","2252033280","784102400","188705792","29786112","2768896","114688"],["21932293","434372048","2928893916","10550793344","23899991440","36841209600","40323713472","32052510720","18672678144","7956930560","2446732288","527400960","75452416","6422528","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","57","387","2538","16245","102105","632772","3877533","23543307","141857838","849234609","5055850161","29955419136","176739279489"],"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":[["156","216","96","0","0"],["-52","292","240","-64","-32"],["0","-52","248","112","-32"],["0","0","-24","48","0"]]},"terms":[{"coeff":{"coeffs":[["832","2400","1968","32","-448","-128"],["-416","412","2680","2032","160","-128"],["0","-572","-392","752","416","0"],["0","0","-192","-192","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["169","728","876","-112","-816","-448","-64"],["0","338","1352","1592","368","-256","-64"],["0","0","156","552","528","96","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180119,"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-4050192c0d17f026"]},{"id":180120,"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-4050192c0d17f026"]},{"id":180121,"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-4050192c0d17f026"]}],"region_exact":{},"incoming_edges":[{"id":"e-4050192c0d17f026","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-a59b31e2ece04346","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":"74b990478dcec63e7b3426b7748c291fe326e07f5126a25060f1b4806b3a3af6","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-820fbf6ab98121e9","role":"sequence","name":"basis_map_A013609 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:     4,      4;\n  2:    13,     32,      12;\n  3:    43,    168,     144,      32;\n  4:   142,    760,    1044,     512,      80;\n  5:   469,   3188,    6060,    4736,    1600,     192;\n  6:  1549,  12768,   31068,   33664,   18000,    4608,     448;\n  7:  5116,  49564,  147132,  204736,  151600,   61056,   12544,   1024;\n  8: 16897, 188120,  659460, 1123072, 1068800,  594816,  191296,  32768,  2304;\n  9: 55807, 701976, 2838564, 5724864, 6677520, 4774464, 2116800, 565248, 82944, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(52*n-156+(52*n^2-292*n-216)*x+(24*n^3-248*n^2-240*n-96)*x^2-(48*n^3+112*n^2-64*n)*x^3+(32*n^2+32*n)*x^4)*P(n,x) = -(416*n-832+(572*n^2-412*n-2400)*x+(192*n^3+392*n^2-2680*n-1968)*x^2+(192*n^3-752*n^2-2032*n-32)*x^3-(416*n^2+160*n-448)*x^4+(128*n+128)*x^5)*P(n-1,x) + (169+(338*n+728)*x+(156*n^2+1352*n+876)*x^2+(552*n^2+1592*n-112)*x^3+(528*n^2+368*n-816)*x^4+(96*n^2-256*n-448)*x^5-(64*n+64)*x^6)*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[(-(416*n - 832 + (572*n^2 - 412*n - 2400)*x + (192*n^3 + 392*n^2 - 2680*n - 1968)*x^2 + (192*n^3 - 752*n^2 - 2032*n - 32)*x^3 - (416*n^2 + 160*n - 448)*x^4 + (128*n + 128)*x^5)*P[n - 1, x] + (169 + (338*n + 728)*x + (156*n^2 + 1352*n + 876)*x^2 + (552*n^2 + 1592*n - 112)*x^3 + (528*n^2 + 368*n - 816)*x^4 + (96*n^2 - 256*n - 448)*x^5 - (64*n + 64)*x^6)*D[P[n - 1, x], x])/(-52*n + 156 - (52*n^2 - 292*n - 216)*x - (24*n^3 - 248*n^2 - 240*n - 96)*x^2 + (48*n^3 + 112*n^2 - 64*n)*x^3 - (32*n^2 + 32*n)*x^4)]];\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] = (-(416*n - 832 + (572*n^2 - 412*n - 2400)*x + (192*n^3 + 392*n^2 - 2680*n - 1968)*x^2 + (192*n^3 - 752*n^2 - 2032*n - 32)*x^3 - (416*n^2 + 160*n - 448)*x^4 + (128*n + 128)*x^5)*v[i - 1] + (169 + (338*n + 728)*x + (156*n^2 + 1352*n + 876)*x^2 + (552*n^2 + 1592*n - 112)*x^3 + (528*n^2 + 368*n - 816)*x^4 + (96*n^2 - 256*n - 448)*x^5 - (64*n + 64)*x^6)*deriv(v[i - 1], x))/(-52*n + 156 - (52*n^2 - 292*n - 216)*x - (24*n^3 - 248*n^2 - 240*n - 96)*x^2 + (48*n^3 + 112*n^2 - 64*n)*x^3 - (32*n^2 + 32*n)*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-d6b7687ba0ff59dd","steps":["basis A013609","\u03b8+1"],"edges":["e-0d5d22b8fd97251b","e-4050192c0d17f026"],"text":"A116412 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A116412 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}