{"id":"n-3b0976fc57c747e3","revision":0,"first_index":0,"fingerprint":"f6-383c3e2053bee1bd","name":"theta_plus_one_squared of Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["0","4"],["0","12","36","16"],["0","36","216","352","200","36"],["0","108","972","2736","3400","2052","588","64"],["0","324","3888","15552","30000","31896","19600","6912","1296","100"],["0","972","14580","75600","198000","303480","291256","179840","71280","17500","2420","144"],["0","2916","52488","334368","1107000","2221020","2914128","2600704","1605744","685500","198440","37152","4056","196"],["0","8748","183708","1388016","5556600","13791708","23004324","26975424","22800528","14049700","6311844","2043216","463736","69972","6300","256"],["0","26244","629856","5505408","25855200","76504176","154992096","225630720","242992224","196893400","120996128","56407680","19798688","5142256","957600","120832","9248","324"],["0","78732","2125764","21100176","113724000","390825648","932175216","1622654208","2127443616","2143189800","1678940824","1028731104","493193376","184051056","52844400","11437056","1803360","195372","12996","400"],["0","236196","7085880","78732000","478953000","1875002580","5147253216","10432903680","16137960480","19470861000","18587894160","14164699968","8653868080","4240320840","1660284000","515205120","124931232","23148180","3162360","300000","17640","484"],["0","708588","23383404","287529264","1948617000","8560924020","26588854908","61523820864","109739181024","154407627000","174109973400","159016660704","118418605552","72146447912","35973135000","14640130560","4833809376","1281746268","268709628","43494000","5239080","441892","23276","576"],["0","2125764","76527504","1031704128","7707862800","37553746824","130195329264","338704084224","684634891248","1101275732700","1433594204064","1528217995392","1344929088672","982228904944","596862021600","301870221312","126816261088","44051029308","12556683216","2903841600","535783248","76953096","8286256","628992","30000","676"],["0","6377292","248714388","3647811024","29784315600","159379864488","610152687144","1763078638464","3983814950544","7213965372900","10653683354028","12994873368048","13211214475680","11265958874352","8091406897200","4904907265024","2510217609120","1082906114004","392406865708","118748401200","29752618896","6096608232","1003986984","129557376","12610000","870012","37908","784"],["0","19131876","803538792","12737577888","112842639000","657904826124","2756570137776","8760314384640","21885650206992","44069866740900","72827773442424","100087784885088","115517292423624","112772870660268","93594559514400","66254335424512","40072337332832","20713384406988","9139106413528","3432365736800","1092305141928","292590061764","65355376944","12016892160","1786330000","209215916","18574920","1174432","47096","900"]],"degrees":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29],"row_sums":["4","64","840","9920","109568","1155072","11763712","116654080","1132462080","10804527104","101602820096","943819063296","8675833937920","79027398246400","714132802240512"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0","4"]],"recurrence":{"denominator":{"coeffs":[["9","9","1","-1","0"],["0","15","17","5","0"],["0","0","4","4","1"]]},"terms":[{"coeff":{"coeffs":[["27","108","171","136","57","12","1"],["0","45","135","154","83","21","2"],["0","0","12","28","23","8","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182823,"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-a5c884a6ac52ea55"]},{"id":182824,"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-a5c884a6ac52ea55"]},{"id":182825,"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-a5c884a6ac52ea55"]}],"region_exact":{},"incoming_edges":[{"id":"e-a5c884a6ac52ea55","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-3b0976fc57c747e3","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":"38de53b6cd6e481de93e78e34b807d343a50494873cad58d55395e2cbb79f176","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-7e353d0275e3ebc5","role":"sequence","name":"Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= 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: 0,     4;\n  1: 0,    12,      36,       16;\n  2: 0,    36,     216,      352,       200,        36;\n  3: 0,   108,     972,     2736,      3400,      2052,       588,         64;\n  4: 0,   324,    3888,    15552,     30000,     31896,     19600,       6912,       1296,        100;\n  5: 0,   972,   14580,    75600,    198000,    303480,    291256,     179840,      71280,      17500,       2420,        144;\n  6: 0,  2916,   52488,   334368,   1107000,   2221020,   2914128,    2600704,    1605744,     685500,     198440,      37152,      4056,       196;\n  7: 0,  8748,  183708,  1388016,   5556600,  13791708,  23004324,   26975424,   22800528,   14049700,    6311844,    2043216,    463736,     69972,     6300,      256;\n  8: 0, 26244,  629856,  5505408,  25855200,  76504176, 154992096,  225630720,  242992224,  196893400,  120996128,   56407680,  19798688,   5142256,   957600,   120832,    9248,    324;\n  9: 0, 78732, 2125764, 21100176, 113724000, 390825648, 932175216, 1622654208, 2127443616, 2143189800, 1678940824, 1028731104, 493193376, 184051056, 52844400, 11437056, 1803360, 195372, 12996, 400;\n","recurrence_sentence":"The row polynomials satisfy (9+(15*n+9)*x+(4*n^2+17*n+1)*x^2+(4*n^2+5*n-1)*x^3+n^2*x^4)*P(n,x) = (27+(45*n+108)*x+(12*n^2+135*n+171)*x^2+(28*n^2+154*n+136)*x^3+(23*n^2+83*n+57)*x^4+(8*n^2+21*n+12)*x^5+(n^2+2*n+1)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 4*x.","mathematica":"P[0, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((27 + (45*n + 108)*x + (12*n^2 + 135*n + 171)*x^2 + (28*n^2 + 154*n + 136)*x^3 + (23*n^2 + 83*n + 57)*x^4 + (8*n^2 + 21*n + 12)*x^5 + (n^2 + 2*n + 1)*x^6)*P[n - 1, x])/(9 + (15*n + 9)*x + (4*n^2 + 17*n + 1)*x^2 + (4*n^2 + 5*n - 1)*x^3 + n^2*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (45*n + 108)*x + (12*n^2 + 135*n + 171)*x^2 + (28*n^2 + 154*n + 136)*x^3 + (23*n^2 + 83*n + 57)*x^4 + (8*n^2 + 21*n + 12)*x^5 + (n^2 + 2*n + 1)*x^6)*v[i - 1])/(9 + (15*n + 9)*x + (4*n^2 + 17*n + 1)*x^2 + (4*n^2 + 5*n - 1)*x^3 + n^2*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-7e353d0275e3ebc5","steps":["(\u03b8+1)\u00b2"],"edges":["e-a5c884a6ac52ea55"],"text":"A299989 \u2192 (\u03b8+1)\u00b2"},"headline":"A299989 \u2192 (\u03b8+1)\u00b2","chain":null}