{"id":"n-d06e1fa708981619","revision":0,"first_index":0,"fingerprint":"f6-e75b70736cdfa66a","name":"theta_plus_one of basis_map_A013610 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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","12"],["8","54","54"],["21","198","405","216"],["55","666","2052","2268","810"],["144","2130","8748","14796","10935","2916"],["377","6588","33777","77112","87480","48114","10206"],["987","19896","122229","351648","540675","456354","199017","34992"],["2584","59022","422442","1466748","2853225","3267378","2184084","787320","118098"],["6765","172674","1410696","5738364","13519305","19617390","17789058","9815256","3011499","393660"],["17711","499590","4586679","21391128","59223555","104254290","119843955","89649504","42042888","11219310","1299078"],["46368","1432362","14597496","76781736","244368090","506350278","706632822","670026816","425684241","173407230","40920957","4251528"],["121393","4075704","45651438","267344064","961812225","2294521668","3772280484","4343172048","3496350339","1927949850","693707652","146677716","13817466"],["317811","11523108","140697486","907856208","3643608825","9839738736","18641034468","25293022416","24721572438","17264943450","8402436504","2706097572","518154975","44641044"],["832040","32400486","428295006","3019014504","13373475030","40341686310","86589908496","135448013376","156032317629","132272121960","81487915245","35470498104","10335464568","1807962282","143489070"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","116","840","5851","39669","263654","1725798","11160901","71474667","454027688","2864499924","17967482047","112135645041","696783394106"],"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":[["150","900","1458","0"],["-60","-684","-1107","648"],["0","0","-405","-972"],["0","0","54","324"]]},"terms":[{"coeff":{"coeffs":[["360","2754","6480","4536","972"],["-90","-981","-2970","-2754","-486"],["0","-45","-999","-2349","-972"],["0","0","81","567","486"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-25","-510","-2997","-6480","-4536","-972"],["0","0","-180","-918","-1296","-486"],["0","0","45","432","1053","486"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181431,"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-9205dd3d97c816ee"]},{"id":181432,"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-9205dd3d97c816ee"]},{"id":181433,"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-9205dd3d97c816ee"]},{"id":181434,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-9205dd3d97c816ee"]}],"region_exact":{},"incoming_edges":[{"id":"e-9205dd3d97c816ee","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-d06e1fa708981619","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":"38f943c976975a6849598fd076458cbacbb69eb529e832e9541713b510aca203","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-de2147e0b2be5082","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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:    3,     12;\n  2:    8,     54,      54;\n  3:   21,    198,     405,     216;\n  4:   55,    666,    2052,    2268,      810;\n  5:  144,   2130,    8748,   14796,    10935,     2916;\n  6:  377,   6588,   33777,   77112,    87480,    48114,    10206;\n  7:  987,  19896,  122229,  351648,   540675,   456354,   199017,   34992;\n  8: 2584,  59022,  422442, 1466748,  2853225,  3267378,  2184084,  787320,  118098;\n  9: 6765, 172674, 1410696, 5738364, 13519305, 19617390, 17789058, 9815256, 3011499, 393660;\n","recurrence_sentence":"The row polynomials satisfy -(60*n-150+(684*n-900)*x-(54*n^3-405*n^2-1107*n+1458)*x^2-(324*n^3-972*n^2+648*n)*x^3)*P(n,x) = -(90*n-360+(45*n^2+981*n-2754)*x-(81*n^3-999*n^2-2970*n+6480)*x^2-(567*n^3-2349*n^2-2754*n+4536)*x^3-(486*n^3-972*n^2-486*n+972)*x^4)*P(n-1,x) - (25+510*x-(45*n^2-180*n-2997)*x^2-(432*n^2-918*n-6480)*x^3-(1053*n^2-1296*n-4536)*x^4-(486*n^2-486*n-972)*x^5)*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[(-(90*n - 360 + (45*n^2 + 981*n - 2754)*x - (81*n^3 - 999*n^2 - 2970*n + 6480)*x^2 - (567*n^3 - 2349*n^2 - 2754*n + 4536)*x^3 - (486*n^3 - 972*n^2 - 486*n + 972)*x^4)*P[n - 1, x] - (25 + 510*x - (45*n^2 - 180*n - 2997)*x^2 - (432*n^2 - 918*n - 6480)*x^3 - (1053*n^2 - 1296*n - 4536)*x^4 - (486*n^2 - 486*n - 972)*x^5)*D[P[n - 1, x], x])/(-60*n + 150 - (684*n - 900)*x + (54*n^3 - 405*n^2 - 1107*n + 1458)*x^2 + (324*n^3 - 972*n^2 + 648*n)*x^3)]];\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] = (-(90*n - 360 + (45*n^2 + 981*n - 2754)*x - (81*n^3 - 999*n^2 - 2970*n + 6480)*x^2 - (567*n^3 - 2349*n^2 - 2754*n + 4536)*x^3 - (486*n^3 - 972*n^2 - 486*n + 972)*x^4)*v[i - 1] - (25 + 510*x - (45*n^2 - 180*n - 2997)*x^2 - (432*n^2 - 918*n - 6480)*x^3 - (1053*n^2 - 1296*n - 4536)*x^4 - (486*n^2 - 486*n - 972)*x^5)*deriv(v[i - 1], x))/(-60*n + 150 - (684*n - 900)*x + (54*n^3 - 405*n^2 - 1107*n + 1458)*x^2 + (324*n^3 - 972*n^2 + 648*n)*x^3)); 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-353fb5a029e4e669","steps":["basis A013610","\u03b8+1"],"edges":["e-4d45de18adc3c05d","e-9205dd3d97c816ee"],"text":"A207607 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A207607 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}