{"id":"n-4220c6692cf7fad9","revision":0,"first_index":1,"fingerprint":"f6-4a484cf9cab203b5","name":"theta_plus_one of basis_map_A013609 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","12"],["16","80","60"],["64","448","672","224"],["256","2304","5184","3840","720"],["1024","11264","33792","39424","17600","2112"],["4096","53248","199680","319488","232960","69888","5824"],["16384","245760","1105920","2252800","2304000","1161216","250880","15360"],["65536","1114112","5849088","14483456","19148800","13787136","5117952","835584","39168"],["262144","4980736","29884416","87162880","141639680","133545984","71909376","20545536","2626560","97280"],["1048576","22020096","148635648","499122176","963379200","1127153664","803618816","337379328","76640256","7884800","236544"],["4194304","96468992","723517440","2749366272","6149898240","8609857536","7681343488","4310958080","1454948352","269434880","22798336","565248"],["16777216","419430400","3460300800","14680064000","37355520000","60964208640","65509785600","46347059200","21086208000","5857280000","902021120","63897600","1331200"],["67108864","1811939328","16307453952","76403441664","217998950400","406667132928","512099352576","438942302208","253402546176","95590809600","22267035648","2898395136","174440448","3096576"],["268435456","7784628224","75900125184","389231411200","1230944337920","2584983109632","3736986451968","3771652374528","2651943075840","1276861480960","406579576832","80643883008","8993374208","465633280","7127040"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","156","1408","12304","105216","885184","7352320","60440832","492654592","3987119104","32073351168","256663883776","2044634005504","16223245025280"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["6","-3"],["-10","11"],["4","-12"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["24","60","24"],["-40","-76","-10"],["16","0","-40"],["0","16","24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","22","12"],["0","-8","-16","0"],["0","0","-8","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183221,"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-68cd1786074e51ab"]},{"id":183222,"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-68cd1786074e51ab"]},{"id":183223,"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-68cd1786074e51ab"]},{"id":183224,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-68cd1786074e51ab"]}],"region_exact":{},"incoming_edges":[{"id":"e-68cd1786074e51ab","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-4220c6692cf7fad9","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":"11f98f85d051709b917f664b77b2bb2eb30bf2401cc39ac617006511c42d9cbb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-504214918830dff1","role":"sequence","name":"basis_map_A013609 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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:      4,      12;\n   3:     16,      80,       60;\n   4:     64,     448,      672,      224;\n   5:    256,    2304,     5184,     3840,       720;\n   6:   1024,   11264,    33792,    39424,     17600,      2112;\n   7:   4096,   53248,   199680,   319488,    232960,     69888,     5824;\n   8:  16384,  245760,  1105920,  2252800,   2304000,   1161216,   250880,    15360;\n   9:  65536, 1114112,  5849088, 14483456,  19148800,  13787136,  5117952,   835584,   39168;\n  10: 262144, 4980736, 29884416, 87162880, 141639680, 133545984, 71909376, 20545536, 2626560, 97280;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-10*n+6+(4*n^3-12*n^2+11*n-3)*x)*P(n,x) = (16*n^2-40*n+24+(16*n^3-76*n+60)*x+(24*n^3-40*n^2-10*n+24)*x^2)*P(n-1,x) - ((8*n-8)*x+(8*n^2+16*n-22)*x^2+(16*n^2-12)*x^3)*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[((16*n^2 - 40*n + 24 + (16*n^3 - 76*n + 60)*x + (24*n^3 - 40*n^2 - 10*n + 24)*x^2)*P[n - 1, x] - ((8*n - 8)*x + (8*n^2 + 16*n - 22)*x^2 + (16*n^2 - 12)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (4*n^3 - 12*n^2 + 11*n - 3)*x)]];\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] = ((16*n^2 - 40*n + 24 + (16*n^3 - 76*n + 60)*x + (24*n^3 - 40*n^2 - 10*n + 24)*x^2)*v[i - 1] - ((8*n - 8)*x + (8*n^2 + 16*n - 22)*x^2 + (16*n^2 - 12)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (4*n^3 - 12*n^2 + 11*n - 3)*x)); 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-ba8de4e02acfb6ca","steps":["basis A013609","\u03b8+1"],"edges":["e-190e8c51191ea68a","e-68cd1786074e51ab"],"text":"A091042 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A091042 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}