{"id":"n-89cce58696f5dce2","revision":0,"first_index":0,"fingerprint":"f6-478b333d0be8428f","name":"theta_plus_one_squared of derivative of Even-numbered rows of Pascal's triangle.","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":[[],["2","8"],["4","48","108","64"],["6","120","540","960","750","216"],["8","224","1512","4480","7000","6048","2744","512"],["10","360","3240","13440","31500","45360","41160","23040","7290","1000"],["12","528","5940","31680","99000","199584","271656","253440","160380","66000","15972","1728"],["14","728","9828","64064","250250","648648","1177176","1537536","1459458","1001000","484484","157248","30758","2744"],["16","960","15120","116480","546000","1729728","3923920","6589440","8339760","8008000","5813808","3144960","1230320","329280","54000","4096"],["18","1224","22032","195840","1071000","4009824","10915632","22404096","35443980","43758000","42357744","32078592","18823896","8396640","2754000","626688","88434","5832"],["20","1520","30780","310080","1938000","8372160","26589360","64496640","122442840","184756000","223554760","217676160","170311440","106357440","52326000","19845120","5600820","1108080","137180","8000"],["22","1848","41580","468160","3291750","16116408","58496592","163722240","362619180","646646000","938929992","1117404288","1092831740","877448880","575586000","305614848","129378942","42661080","10562860","1848000","203742","10648"],["24","2208","54648","680064","5313000","29072736","118713672","376561152","953170416","1961256000","3322367664","4672781568","5484028368","5381686464","4412826000","3012489216","1700408952","784963872","291534936","85008000","18744264","2938848","292008","13824"],["26","2600","70200","956800","8222500","49729680","225625400","799884800","2277796950","5311735000","10283518960","16688505600","22850118200","26500728800","26075790000","21756866560","15350914150","9111187800","4511850200","1841840000","609188580","159187600","31634200","4492800","406250","17576"],["28","3024","88452","1310400","12285000","81375840","406125720","1591349760","5035130100","13123110000","28582133580","52568792640","82260425520","110079950400","126367290000","124607508480","105502646340","76533977520","47374427100","24864840000","10965394440","4011527520","1195772760","283046400","51187500","6643728","551124","21952"]],"degrees":[-1,1,3,5,7,9,11,13,15,17,19,21,23,25,27],"row_sums":["0","10","224","2592","22528","166400","1105920","6823936","39845888","222953472","1205862400","6343884800","32614907904","164450271232","815506915328"],"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"],["2","8"]],"recurrence":{"denominator":{"coeffs":[["-1","7","-4"],["1","-13","12"],["0","6","-12"],["0","0","4"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0","0"],["1","1","-1","-1","0"],["0","6","12","6","0"],["0","0","4","8","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177696,"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-73f121b40ecb8361"]},{"id":177697,"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-73f121b40ecb8361"]},{"id":177698,"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-73f121b40ecb8361"]}],"region_exact":{},"incoming_edges":[{"id":"e-73f121b40ecb8361","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-89cce58696f5dce2","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":"ab0f2ad08fc7810a12b130b0b60aff181db34a01e94c0d2c408faaa399f5e32a","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-f1591709bf5cf41b","role":"sequence","name":"derivative of Even-numbered rows of Pascal's triangle.","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;\n  1:  2,    8;\n  2:  4,   48,   108,     64;\n  3:  6,  120,   540,    960,     750,     216;\n  4:  8,  224,  1512,   4480,    7000,    6048,     2744,      512;\n  5: 10,  360,  3240,  13440,   31500,   45360,    41160,    23040,     7290,     1000;\n  6: 12,  528,  5940,  31680,   99000,  199584,   271656,   253440,   160380,    66000,    15972,     1728;\n  7: 14,  728,  9828,  64064,  250250,  648648,  1177176,  1537536,  1459458,  1001000,   484484,   157248,    30758,    2744;\n  8: 16,  960, 15120, 116480,  546000, 1729728,  3923920,  6589440,  8339760,  8008000,  5813808,  3144960,  1230320,  329280,   54000,   4096;\n  9: 18, 1224, 22032, 195840, 1071000, 4009824, 10915632, 22404096, 35443980, 43758000, 42357744, 32078592, 18823896, 8396640, 2754000, 626688, 88434, 5832;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(6*n^2-13*n+7)*x+(4*n^3-12*n^2+12*n-4)*x^2)*P(n,x) = (n+(6*n^2+n)*x+(4*n^3+12*n^2-n)*x^2+(8*n^3+6*n^2-n)*x^3+4*n^3*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (6*n^2 + n)*x + (4*n^3 + 12*n^2 - n)*x^2 + (8*n^3 + 6*n^2 - n)*x^3 + 4*n^3*x^4)*P[n - 1, x])/(n - 1 + (6*n^2 - 13*n + 7)*x + (4*n^3 - 12*n^2 + 12*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + (6*n^2 + n)*x + (4*n^3 + 12*n^2 - n)*x^2 + (8*n^3 + 6*n^2 - n)*x^3 + 4*n^3*x^4)*v[i - 1])/(n - 1 + (6*n^2 - 13*n + 7)*x + (4*n^3 - 12*n^2 + 12*n - 4)*x^2)); 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-08f49539099949d5","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-473cc94657c879cc","e-73f121b40ecb8361"],"text":"A034870 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A034870 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}