{"id":"n-2769916898bf94a3","revision":0,"first_index":0,"fingerprint":"f6-5b73cb7ea46d8e57","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["18","11"],["203","396","121"],["1960","6699","6534","1331"],["17493","86240","147378","95832","14641"],["148862","962115","2371600","2701930","1317690","161051"],["1226911","9824892","31749795","52175200","44581845","17393508","1771561"],["9882516","94472147","378258342","814911405","1004372600","686560413","223216686","19487171"],["78236585","869661408","4156774468","11095578032","17928050910","17676957760","10069552724","2806152624","214358881"],["611068906","7745421915","43048239696","137173557444","274615556292","354975408018","291669803040","142412245668","34726138722","2357947691"],["4721372019","67217579660","425998205325","1578435455520","3772272829710","6041542238424","6507882480330","4583382619200","1958168377935","424430584380","25937424601"],["36156831872","571286014299","4066663569430","17181927614775","47747672529480","91289002478982","121837768474884","112493397159990","69323662115400","26326485970015","5135610070998","285311670611"],["274848417277","4772701807104","37704876943734","178933197054920","567003611287575","1260538554778272","2008358054537604","2297512205526384","1856141053139835","1016747044359200","347509614804198","61627320851976","3138428376721"],["2076193080150","39303323670611","341248179207936","1797265800984654","6396861794713390","16216303282824645","30042835555548816","41027885971268196","41068030673784114","29492018955444045","14539482734336560","4517624992454574","734392240152714","34522712143931"],["15599130675527","319733734343100","3026355922637047","17517406532674048","69194733337909179","197023343277172412","416218450925832555","660942382222073952","789786804946912773","702719635973639284","454177091913838293","203552758280711840","57976187403167033","8699723460270612","379749833583241"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","29","720","16524","361584","7663248","158723712","3231161280","64895323392","1289335387392","25389989167104","496294944500736","9640261511884800","186249852409614336","3581549956895440896"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-4","11"],["11","0"]]},"terms":[{"coeff":{"coeffs":[["49","154","121"],["77","121","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189827,"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-fc274cdac51235f1"]},{"id":189828,"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-fc274cdac51235f1"]},{"id":189829,"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-fc274cdac51235f1"]},{"id":189830,"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-fc274cdac51235f1"]}],"region_exact":{},"incoming_edges":[{"id":"e-fc274cdac51235f1","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-2769916898bf94a3","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":"0eb7519514dbe9fd0668c98f2f4a8fe8a2def90d7d33d77cca0b98c08b01dffe","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-b7cf68162d40a956","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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:         1;\n  1:        18,         11;\n  2:       203,        396,         121;\n  3:      1960,       6699,        6534,         1331;\n  4:     17493,      86240,      147378,        95832,        14641;\n  5:    148862,     962115,     2371600,      2701930,      1317690,       161051;\n  6:   1226911,    9824892,    31749795,     52175200,     44581845,     17393508,      1771561;\n  7:   9882516,   94472147,   378258342,    814911405,   1004372600,    686560413,    223216686,     19487171;\n  8:  78236585,  869661408,  4156774468,  11095578032,  17928050910,  17676957760,  10069552724,   2806152624,   214358881;\n  9: 611068906, 7745421915, 43048239696, 137173557444, 274615556292, 354975408018, 291669803040, 142412245668, 34726138722, 2357947691;\n","recurrence_sentence":"The row polynomials satisfy (11*n-4+11*x)*P(n,x) = (77*n+49+(121*n+154)*x+121*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((77*n + 49 + (121*n + 154)*x + 121*x^2)*P[n - 1, x])/(11*n - 4 + 11*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((77*n + 49 + (121*n + 154)*x + 121*x^2)*v[i - 1])/(11*n - 4 + 11*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-b7cf68162d40a956","steps":["D+1"],"edges":["e-fc274cdac51235f1"],"text":"A038277 \u2192 D+1"},"headline":"A038277 \u2192 D+1","chain":null}