{"id":"n-85048335272aedcf","revision":0,"first_index":0,"fingerprint":"f6-86b851ff29f26c39","name":"theta_plus_one_squared of basis_map_A013611 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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","16"],["18","144","144"],["162","1584","2592","1024"],["1944","21600","45360","30720","6400"],["29160","355104","874800","783360","288000","36864"],["524880","6858432","18942336","20321280","10080000","2322432","200704"],["11022480","152425152","459514944","561447936","338688000","106831872","16859136","1048576"],["264539520","3834563328","12400185024","16742412288","11637561600","4514807808","986259456","113246208","5308416"],["7142567040","107765842176","369316065600","540224225280","418854240000","188932165632","51209625600","8210350080","716636160","26214400"],["214277011200","3347256337920","12049374547584","18832974336000","15942028608000","8080565391360","2564919447552","513854668800","63063982080","4325376000","126877696"],["7071141369600","113887891330560","427754667111552","707172594315264","643793033664000","358484742696960","128636531122176","30357599551488","4682500669440","454164480000","25121783808","603979776"],["254561089305600","4213102349813760","16424159037990912","28500024361476096","27596377926374400","16613698610995200","6582665386192896","1764933828083712","322255371829248","39473381376000","3102540300288","141331267584","2835349504"],["9927882482918400","168383969071626240","678460123629969408","1228294969923280896","1254383891387827200","806889382684729344","347176309165830144","103223487435374592","21502937006014464","3130846425907200","312050237571072","20281036898304","774050414592","13153337344"],["416971064282572800","7230972820734996480","30010780914103351296","56412994060368691200","60360967000309248000","41114605287152517120","18974469401803948032","6149205556907212800","1425692259394191360","237682893127680000","28259406679375872","2337265837670400","127718318407680","4143301263360","60397977600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","19","306","5362","106024","2367288","59250064","1647838096","50498883648","1692397931968","61612766584192","2422320592074624","102314498050355200","4621706917691699200","222365069852719104000"],"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"],["3","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","48","144"],["18","-48","0"],["-9","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-48","64"],["0","0","48","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","16"],["6","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189155,"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-cc86f69189f44c47"]},{"id":189156,"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-cc86f69189f44c47"]},{"id":189157,"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-cc86f69189f44c47"]},{"id":189158,"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-cc86f69189f44c47"]}],"region_exact":{},"incoming_edges":[{"id":"e-cc86f69189f44c47","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-85048335272aedcf","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":"7bf037ac14561ef8ced20897ca8c1c81a61b80828e97b5cc767ea27737dc5505","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-7ba399ade8b53e3c","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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,           16;\n  2:         18,          144,          144;\n  3:        162,         1584,         2592,         1024;\n  4:       1944,        21600,        45360,        30720,         6400;\n  5:      29160,       355104,       874800,       783360,       288000,        36864;\n  6:     524880,      6858432,     18942336,     20321280,     10080000,      2322432,      200704;\n  7:   11022480,    152425152,    459514944,    561447936,    338688000,    106831872,    16859136,    1048576;\n  8:  264539520,   3834563328,  12400185024,  16742412288,  11637561600,   4514807808,   986259456,  113246208,   5308416;\n  9: 7142567040, 107765842176, 369316065600, 540224225280, 418854240000, 188932165632, 51209625600, 8210350080, 716636160, 26214400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(48*n-48)*x-144*x^2)*P(n-2,x) + ((48*n-48)*x^2+64*x^3)*d/dx P(n-2,x) + (6*n-3+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 16*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (48*n - 48)*x - 144*x^2)*P[n - 2, x] + ((48*n - 48)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (6*n - 3 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (48*n - 48)*x - 144*x^2)*v[i - 2] + ((48*n - 48)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (6*n - 3 + 16*x)*v[i - 1] - 16*x^2*deriv(v[i - 1], 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-8229dbdd6a4b5cd9","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-6ce7f219250b8b1f","e-cc86f69189f44c47"],"text":"A225470 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A225470 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}