{"id":"n-ce8c9c3ad179c5b9","revision":0,"first_index":0,"fingerprint":"f6-3588fe420a4e2f99","name":"theta_plus_one_squared of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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":[["169","1248","1296"],["2197","24336","50544","27648"],["28561","421824","1314144","1437696","518400"],["371293","6854640","28473120","46725120","33696000","8957952"],["4826809","106932384","555225840","1214853120","1314144000","698720256","146313216"],["62748517","1621807824","10105110288","27637908480","39862368000","31791771648","13314502656","2293235712"],["815730721","24095430528","175155244992","574868496384","1036421568000","1102114750464","692354138112","238496514048","34828517376"],["10604499373","352395671472","2927594809152","11209935679488","24252264691200","32236856451072","27001811386368","13952046071808","4074936532992","515978035200"],["137858491849","5090159699040","47573415648720","208184519761920","525465734976000","838158267727872","877558870056960","604588663111680","264870874644480","67077144576000","7492001071104"],["1792160394037","72789283696272","755888715307440","3721298290744320","10734514300224000","19976105380847616","25098183683629056","21614044706242560","12625511691386880","4796015837184000","1071356153167872","106993205379072"],["23298085122481","1032284386965312","11791863958796064","64502503706234880","209323028854368000","445181777058889728","652552775774355456","674358194834767872","492394955964088320","249392823533568000","83565779947094016","16690940039135232","1506820975755264"],["302875106592253","14538005116428144","181165909912412256","1090092312635369472","3930621319598688000","9404465040369045504","15754488443695153152","18994422487845961728","16642949511586185216","10536846794293248000","4707538937019629568","1410384433306927104","254652744902639616","20970668254298112"],["3937376385699289","203532071629994016","2747682967004919216","18036072809057931264","71537308016696121600","190179181927462920192","358414612094064734208","493854984683995004928","504836135184780951552","383541223312274227200","214193021634393145344","85563322287286910976","23173399786140205056","3816661622282256384","288881654523494400"],["51185893014090757","2834910997703488080","41215244505073788240","293086183147191383040","1268161369386885792000","3708494047585526943744","7765649928704735907840","12037715251672378245120","14063292337290326507520","12465089757648912384000","8353527843741332668416","4171211961505236910080","1506270986099113328640","372124508172519997440","56331922632081408000","3944197523094110208"],["665416609183179841","39310765834821701376","612340775503953425280","4689378930355062128640","21981463736039353728000","70124251081617236754432","161525518517058506883072","278204974705317186109440","365645600769548489195520","370391238512996253696000","289588965249699532505088","173522417598617855459328","78326091277153893089280","25800632566628053155840","5858519953736466432000","820393084803574923264","53431550820665524224"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2713","104725","3720625","125078125","4041015625","126689453125","3879150390625","116534423828125","3446197509765625","100574493408203125","2902317047119140625","82943439483642578125","2350389957427978515625","66109001636505126953125","1847185194492340087890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["169","1248","1296"]],"recurrence":{"denominator":{"coeffs":[["169","780","576"],["0","468","576"],["0","0","144"]]},"terms":[{"coeff":{"coeffs":[["2197","18252","31824","15552"],["0","6084","16848","10368"],["0","0","1872","1728"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194017,"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-9dd814c9d2d83b42"]},{"id":194018,"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-9dd814c9d2d83b42"]},{"id":194019,"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-9dd814c9d2d83b42"]},{"id":194020,"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-9dd814c9d2d83b42"]}],"region_exact":{},"incoming_edges":[{"id":"e-9dd814c9d2d83b42","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-ce8c9c3ad179c5b9","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":"bb07934d990d7d62f405b9b7efe3ae5dcd6b351eb08f5251d15572ebe98690c9","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-b8a3f88eab200094","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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:           169,           1248,            1296;\n  1:          2197,          24336,           50544,            27648;\n  2:         28561,         421824,         1314144,          1437696,            518400;\n  3:        371293,        6854640,        28473120,         46725120,          33696000,           8957952;\n  4:       4826809,      106932384,       555225840,       1214853120,        1314144000,         698720256,         146313216;\n  5:      62748517,     1621807824,     10105110288,      27637908480,       39862368000,       31791771648,       13314502656,        2293235712;\n  6:     815730721,    24095430528,    175155244992,     574868496384,     1036421568000,     1102114750464,      692354138112,      238496514048,       34828517376;\n  7:   10604499373,   352395671472,   2927594809152,   11209935679488,    24252264691200,    32236856451072,    27001811386368,    13952046071808,     4074936532992,     515978035200;\n  8:  137858491849,  5090159699040,  47573415648720,  208184519761920,   525465734976000,   838158267727872,   877558870056960,   604588663111680,   264870874644480,   67077144576000,    7492001071104;\n  9: 1792160394037, 72789283696272, 755888715307440, 3721298290744320, 10734514300224000, 19976105380847616, 25098183683629056, 21614044706242560, 12625511691386880, 4796015837184000, 1071356153167872, 106993205379072;\n","recurrence_sentence":"The row polynomials satisfy (169+(468*n+780)*x+(144*n^2+576*n+576)*x^2)*P(n,x) = (2197+(6084*n+18252)*x+(1872*n^2+16848*n+31824)*x^2+(1728*n^2+10368*n+15552)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 1248*x + 1296*x^2.","mathematica":"P[0, x_] := 169 + 1248*x + 1296*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2197 + (6084*n + 18252)*x + (1872*n^2 + 16848*n + 31824)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*P[n - 1, x])/(169 + (468*n + 780)*x + (144*n^2 + 576*n + 576)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 169 + 1248*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (6084*n + 18252)*x + (1872*n^2 + 16848*n + 31824)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*v[i - 1])/(169 + (468*n + 780)*x + (144*n^2 + 576*n + 576)*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-4397e742863fdbe2","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-7b93be005ec5b750","e-9dd814c9d2d83b42"],"text":"A038305 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038305 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}