{"id":"n-d3760c0acdc98316","revision":0,"first_index":0,"fingerprint":"f6-85b44f7305edc983","name":"one_plus_derivative 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":true,"rows":[["481","600","144"],["8281","17316","10800","1728"],["134017","397488","415584","172800","20736"],["2084953","8041020","11924640","8311680","2592000","248832"],["31559905","150116616","289476720","286191360","149610240","37324800","2985984"],["468200473","2651032020","6304897872","8105348160","6010018560","2513452032","522547200","35831808"],["6839588353","44947245408","127249536960","201756731904","194528355840","115392356352","40215232512","7166361600","429981696"],["98703417241","738675542124","2427151252032","4580983330560","5447431761408","4201812486144","2077062414336","620463587328","96745881600","5159780352"],["1410398416609","11844410068920","44320532527440","97086050081280","137429499916800","130738362273792","84036249722880","35606784245760","9306953809920","1289945088000","61917364224"],["19989481318105","186172590992388","781731064548720","1950103431207360","3203839652682240","3628138797803520","2876243970023424","1584683566202880","587511940055040","136501989212160","17027275161600","743008370688"],["281369181863809","2878485309807120","13404426551451936","37523091098338560","70203723523464960","92270581997248512","87075331147284480","59168447383339008","28524304191651840","9400191040880640","1965628644655104","222902511206400","8916100448256"],["3937376385699289","43893592370754204","224521854164955360","697030180675500672","1463400552835203840","2190356173932106752","2399035131928461312","1940535951282339840","1153784723975110656","494421272655298560","146642980237737984","27876188051472384","2897732645683200","106993205379072"],["54820394293197793","661479232797480552","3687061759143353136","12573223833237500160","29275267588371028224","49170258575262849024","61329972870098989056","57576843166283071488","40751254976929136640","21537314847535398912","8306277380609015808","2239638243630907392","390266632720613376","37447621882675200","1283918464548864"],["759913642439962777","9867670972775602740","59533130951773249680","221223705548601188160","565795072495687507200","1053909633181357016064","1475107757257885470720","1577056445231116861440","1295478971241369108480","815025099538582732800","387671667255637180416","135920902591783895040","33594573654463610880","5403691837670031360","481469424205824000","15407021574586368"],["10493108067888605185","145903419348472853184","947296413386457863040","3810120380913487979520","10618737866332857031680","21726530783834400276480","33725108261803424514048","40460098484787715768320","37849354685546804674560","27636884719815874314240","15648481911140788469760","6766632737552939876352","2174734441468542320640","496166010896693329920","74107773773760430080","6162808629834547200","184884258895036416"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1225","38125","1140625","33203125","947265625","26611328125","738525390625","20294189453125","553131103515625","14972686767578125","402927398681640625","10788440704345703125","287592411041259765625","7636845111846923828125","202096998691558837890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["481","600","144"]],"recurrence":{"denominator":{"coeffs":[["25","12"],["12","0"]]},"terms":[{"coeff":{"coeffs":[["481","600","144"],["156","144","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194009,"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-ac7a45e6c74791ca"]},{"id":194010,"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-ac7a45e6c74791ca"]},{"id":194011,"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-ac7a45e6c74791ca"]},{"id":194012,"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-ac7a45e6c74791ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-ac7a45e6c74791ca","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-d3760c0acdc98316","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"3c7c89e3745107c2686cb4ba9a0b96aceb896afd5444708a2b2185aff0c910a0","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:            481,             600,             144;\n  1:           8281,           17316,           10800,             1728;\n  2:         134017,          397488,          415584,           172800,            20736;\n  3:        2084953,         8041020,        11924640,          8311680,          2592000,           248832;\n  4:       31559905,       150116616,       289476720,        286191360,        149610240,         37324800,          2985984;\n  5:      468200473,      2651032020,      6304897872,       8105348160,       6010018560,       2513452032,        522547200,         35831808;\n  6:     6839588353,     44947245408,    127249536960,     201756731904,     194528355840,     115392356352,      40215232512,       7166361600,       429981696;\n  7:    98703417241,    738675542124,   2427151252032,    4580983330560,    5447431761408,    4201812486144,    2077062414336,     620463587328,     96745881600,      5159780352;\n  8:  1410398416609,  11844410068920,  44320532527440,   97086050081280,  137429499916800,  130738362273792,   84036249722880,   35606784245760,   9306953809920,   1289945088000,    61917364224;\n  9: 19989481318105, 186172590992388, 781731064548720, 1950103431207360, 3203839652682240, 3628138797803520, 2876243970023424, 1584683566202880, 587511940055040, 136501989212160, 17027275161600, 743008370688;\n","recurrence_sentence":"The row polynomials satisfy (12*n+25+12*x)*P(n,x) = (156*n+481+(144*n+600)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 481 + 600*x + 144*x^2.","mathematica":"P[0, x_] := 481 + 600*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((156*n + 481 + (144*n + 600)*x + 144*x^2)*P[n - 1, x])/(12*n + 25 + 12*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 481 + 600*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((156*n + 481 + (144*n + 600)*x + 144*x^2)*v[i - 1])/(12*n + 25 + 12*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-4397e742863fdbe2","steps":["basis A013611","D+1"],"edges":["e-7b93be005ec5b750","e-ac7a45e6c74791ca"],"text":"A038305 \u2192 basis A013611 \u2192 D+1"},"headline":"A038305 \u2192 basis A013611 \u2192 D+1","chain":null}