{"id":"n-6ed674fe0b3aea00","revision":0,"first_index":0,"fingerprint":"f6-576b9a231d8a9de1","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["144","672","441"],["1728","12096","15876","5488"],["20736","193536","381024","263424","60025"],["248832","2903040","7620480","7902720","3601500","605052"],["2985984","41803776","137168640","189665280","129654000","43563744","5764801"],["35831808","585252864","2304433152","3982970880","3630312000","1829677248","484243284","52706752"],["429981696","8026324992","36870930432","76473040896","87127488000","58549671936","23243677632","5059848192","466948881"],["5159780352","108355387392","568865783808","1376514736128","1881953740800","1580841142272","836772394752","273231802368","50430479148","4035360700"],["61917364224","1444738498560","8532986757120","23597395476480","37639074816000","37940187414528","25103171842560","10929272094720","3025828748880","484243284000","34179505129"],["743008370688","19070548180992","125150472437760","389357025361920","709765410816000","834684123119616","662723736643584","360665979125760","133136464950720","31960056744000","4511694677028","284735050992"],["8916100448256","249650812551168","1802166803103744","6229712405790720","12775777394688000","17170644818460672","15905369679446016","10387180198821888","4792912738225920","1534082723712000","324842016746016","41001847342848","2339177536969"],["106993205379072","3245460563165184","25558001934925824","97183513530335232","221446808174592000","334827573959983104","354462524284796928","270066685169369088","149538877432648704","59829226224768000","16891784870792832","3198144092742144","364911695767164","18990246039772"],["1283918464548864","41941336508596224","357812027088961536","1484257297554210816","3720306377333145600","6250114713919684608","7443713009980735488","6481600444064858112","4187088568114163712","2010262001152204800","709454964573298944","179096069193560064","30652582444441776","3190361334681696","152600191391025"],["15407021574586368","539245755110522880","4954320375077928960","22263859463313162240","60877740719996928000","112502064850554322944","148874260199614709760","145836009991459307520","107667991751507066880","60307860034566144000","25540378724638761984","8059323113710202880","1839154946666506560","287132520121352640","27468034450384500","1215375746545408"],["184884258895036416","6902345665414692864","67944965143925882880","328820078227394396160","974043851519950848000","1963672404664220909568","2858385795832602427392","3111168213151131893760","2584031802036169605120","1654158446662385664000","817292119188440383488","309478007566471790592","88279437439992314880","18376481287766568960","2636931307236912000","233352143336718336","9604316934614689"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1257","35188","918745","22881624","550606225","12905427988","296247912657","6686160607720","148792995802201","3272053255479060","71224596716874217","1536739495385305048","32900926271918483265","699593432878108433524","14785618720412270659105"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","672","441"]],"recurrence":{"denominator":{"coeffs":[["144","420","196"],["0","252","196"],["0","0","49"]]},"terms":[{"coeff":{"coeffs":[["1728","9072","9996","3087"],["0","3024","5292","2058"],["0","0","588","343"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193026,"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-7cf88883c9db363a"]},{"id":193027,"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-7cf88883c9db363a"]},{"id":193028,"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-7cf88883c9db363a"]},{"id":193029,"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-7cf88883c9db363a"]}],"region_exact":{},"incoming_edges":[{"id":"e-7cf88883c9db363a","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-6ed674fe0b3aea00","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":"af05a23c4f6c6977caf289ac70f89456edfee7d0def965f202e4c51de8017b36","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-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","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:          144,            672,             441;\n  1:         1728,          12096,           15876,            5488;\n  2:        20736,         193536,          381024,          263424,           60025;\n  3:       248832,        2903040,         7620480,         7902720,         3601500,          605052;\n  4:      2985984,       41803776,       137168640,       189665280,       129654000,        43563744,         5764801;\n  5:     35831808,      585252864,      2304433152,      3982970880,      3630312000,      1829677248,       484243284,        52706752;\n  6:    429981696,     8026324992,     36870930432,     76473040896,     87127488000,     58549671936,     23243677632,      5059848192,       466948881;\n  7:   5159780352,   108355387392,    568865783808,   1376514736128,   1881953740800,   1580841142272,    836772394752,    273231802368,     50430479148,     4035360700;\n  8:  61917364224,  1444738498560,   8532986757120,  23597395476480,  37639074816000,  37940187414528,  25103171842560,  10929272094720,   3025828748880,   484243284000,   34179505129;\n  9: 743008370688, 19070548180992, 125150472437760, 389357025361920, 709765410816000, 834684123119616, 662723736643584, 360665979125760, 133136464950720, 31960056744000, 4511694677028, 284735050992;\n","recurrence_sentence":"The row polynomials satisfy (144+(252*n+420)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (1728+(3024*n+9072)*x+(588*n^2+5292*n+9996)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 672*x + 441*x^2.","mathematica":"P[0, x_] := 144 + 672*x + 441*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1728 + (3024*n + 9072)*x + (588*n^2 + 5292*n + 9996)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(144 + (252*n + 420)*x + (49*n^2 + 196*n + 196)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 672*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1728 + (3024*n + 9072)*x + (588*n^2 + 5292*n + 9996)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(144 + (252*n + 420)*x + (49*n^2 + 196*n + 196)*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-5cf6acfdac607d6d","steps":["(\u03b8+1)\u00b2"],"edges":["e-7cf88883c9db363a"],"text":"A038333 \u2192 (\u03b8+1)\u00b2"},"headline":"A038333 \u2192 (\u03b8+1)\u00b2","chain":null}