{"id":"n-48447a2879d0dd14","revision":0,"first_index":0,"fingerprint":"f6-66f70a0cb8d78fee","name":"theta_plus_one of Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","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":false,"rows":[["1"],["3","4"],["13","36","18"],["73","312","324","96"],["501","2920","4680","2880","600"],["4051","30060","65700","62400","27000","4320"],["37633","340284","946890","1226400","819000","272160","35280"],["394353","4214896","14291928","23567040","21462000","11007360","2963520","322560"],["4596553","56786832","227604384","457341696","530258400","370863360","154103040","34836480","3265920"],["58941091","827379540","3833111160","9104175360","12862735200","11453581440","6490108800","2264371200","440899200","36288000"],["824073141","12967040020","68258812050","187396545600","312956028000","339576209280","244979380800","116557056000","35026992000","5987520000","439084800"],["12470162233","217555309224","1283736961980","4004516973600","7730107506000","9914446967040","8715789371520","5279555635200","2163590352000","570810240000","86938790400","5748019200"],["202976401213","3890690616696","25453971179208","89005762697280","195220202463000","289415225024640","300738224666880","221985819095040","115820251747200","41669147520000","9795103718400","1345036492800","80951270400"],["3535017524403","73883410041532","531079269179004","2058943446495936","5062202753407800","8527218443583840","10242083241149760","8936221532958720","5681449557463680","2602380965184000","834216333350400","176797574553600","22099696819200","1220496076800"],["65573803186921","1484707360249260","11636637081541290","49567398456707040","135118163676295800","255135018771753120","348194753113006800","351157139696563200","263897792145187200","147296840378688000","60115000295750400","17373761653248000","3351787350912000","384456264192000","19615115520000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","67","805","11581","193531","3677647","78223657","1839656665","47371590991","1324968741691","39985266288397","1294623362892757","44753331737788675","1644798645162801031"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","3","0"],["0","2","1"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["2","3","0","0"],["4","15","13","1"],["0","5","11","5"],["0","0","2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-5","-8","-5","-1"],["0","-2","-7","-8","-3"],["0","0","-1","-2","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186824,"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-58a4dbe06c970aa4"]},{"id":186825,"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-58a4dbe06c970aa4"]},{"id":186826,"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-58a4dbe06c970aa4"]}],"region_exact":{},"incoming_edges":[{"id":"e-58a4dbe06c970aa4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-48447a2879d0dd14","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":"7dee0bc499da21fcf056a3dd881f796d7a9f9193284af379122d87fdc1fcc974","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-eb51f16fedb3c293","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","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:        3,         4;\n  2:       13,        36,         18;\n  3:       73,       312,        324,         96;\n  4:      501,      2920,       4680,       2880,         600;\n  5:     4051,     30060,      65700,      62400,       27000,        4320;\n  6:    37633,    340284,     946890,    1226400,      819000,      272160,      35280;\n  7:   394353,   4214896,   14291928,   23567040,    21462000,    11007360,    2963520,     322560;\n  8:  4596553,  56786832,  227604384,  457341696,   530258400,   370863360,  154103040,   34836480,   3265920;\n  9: 58941091, 827379540, 3833111160, 9104175360, 12862735200, 11453581440, 6490108800, 2264371200, 440899200, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (2+(2*n+3)*x+(n^2+n)*x^2)*P(n,x) = (4*n+2+(5*n^2+15*n+3)*x+(2*n^3+11*n^2+13*n)*x^2+(2*n^3+5*n^2+n)*x^3)*P(n-1,x) - (1+(2*n+5)*x+(n^2+7*n+8)*x^2+(2*n^2+8*n+5)*x^3+(n^2+3*n+1)*x^4)*d/dx 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[((4*n + 2 + (5*n^2 + 15*n + 3)*x + (2*n^3 + 11*n^2 + 13*n)*x^2 + (2*n^3 + 5*n^2 + n)*x^3)*P[n - 1, x] - (1 + (2*n + 5)*x + (n^2 + 7*n + 8)*x^2 + (2*n^2 + 8*n + 5)*x^3 + (n^2 + 3*n + 1)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 3)*x + (n^2 + n)*x^2)]];\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] = ((4*n + 2 + (5*n^2 + 15*n + 3)*x + (2*n^3 + 11*n^2 + 13*n)*x^2 + (2*n^3 + 5*n^2 + n)*x^3)*v[i - 1] - (1 + (2*n + 5)*x + (n^2 + 7*n + 8)*x^2 + (2*n^2 + 8*n + 5)*x^3 + (n^2 + 3*n + 1)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 3)*x + (n^2 + n)*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-d55979c30b97975a","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-be016f00370af8fd","e-58a4dbe06c970aa4"],"text":"A089231 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A089231 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}