{"id":"n-b133acdbf6c96265","revision":0,"first_index":0,"fingerprint":"f6-27dc70b4eb5f0f21","name":"theta_plus_one of basis_map_A013609 of Square the entries of Pascal's triangle.","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"],["2","4"],["6","24","12"],["20","120","144","32"],["70","560","1080","640","80"],["252","2520","6720","6720","2400","192"],["924","11088","37800","53760","33600","8064","448"],["3432","48048","199584","369600","336000","145152","25088","1024"],["12870","205920","1009008","2306304","2772000","1774080","564480","73728","2304"],["48620","875160","4942080","13453440","20180160","17297280","8279040","2027520","207360","5120"],["184756","3695120","23629320","74680320","134534400","145297152","94174080","35143680","6842880","563200","11264"],["705432","15519504","110853600","399072960","840153600","1097800704","904071168","461260800","138378240","21964800","1486848","24576"],["2704156","64899744","512143632","2069267200","4988412000","7662200832","7684604928","5018517504","2075673600","512512000","67651584","3833856","53248"],["10400600","270415600","2336390784","10470492032","28452424000","50283192960","59594895360","47675916288","25406244864","8712704000","1804042240","201277440","9691136","114688"],["40116600","1123264800","10546208400","51919795200","157057380480","314114760960","430200650880","408650711040","268177029120","119189790720","34502307840","6083051520","581468160","24084480","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","42","316","2430","18804","145684","1127928","8720694","67315780","518756172","3991292232","30662474284","235228201992","1802210865960"],"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":[["0","0"],["1","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["-2","-8","-6"],["4","10","4"],["0","4","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-7","-6"],["0","0","-2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181502,"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-b2171f799212f32b"]},{"id":181503,"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-b2171f799212f32b"]},{"id":181504,"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-b2171f799212f32b"]},{"id":181505,"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-b2171f799212f32b"]}],"region_exact":{},"incoming_edges":[{"id":"e-b2171f799212f32b","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-b133acdbf6c96265","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":"98801b20d365854cd2ee93f4de30bc3b2d8ce0d025063461797a47fd3271d12c","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-fe801e460afa1e3b","role":"sequence","name":"basis_map_A013609 of Square the entries of Pascal's triangle.","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:     2,      4;\n  2:     6,     24,      12;\n  3:    20,    120,     144,       32;\n  4:    70,    560,    1080,      640,       80;\n  5:   252,   2520,    6720,     6720,     2400,      192;\n  6:   924,  11088,   37800,    53760,    33600,     8064,     448;\n  7:  3432,  48048,  199584,   369600,   336000,   145152,   25088,    1024;\n  8: 12870, 205920, 1009008,  2306304,  2772000,  1774080,  564480,   73728,   2304;\n  9: 48620, 875160, 4942080, 13453440, 20180160, 17297280, 8279040, 2027520, 207360, 5120;\n","recurrence_sentence":"The row polynomials satisfy (n+n^2*x)*P(n,x) = (4*n-2+(4*n^2+10*n-8)*x+(6*n^2+4*n-6)*x^2)*P(n-1,x) - (2*x+(2*n+7)*x^2+(4*n+6)*x^3)*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 + (4*n^2 + 10*n - 8)*x + (6*n^2 + 4*n - 6)*x^2)*P[n - 1, x] - (2*x + (2*n + 7)*x^2 + (4*n + 6)*x^3)*D[P[n - 1, x], x])/(n + n^2*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] = ((4*n - 2 + (4*n^2 + 10*n - 8)*x + (6*n^2 + 4*n - 6)*x^2)*v[i - 1] - (2*x + (2*n + 7)*x^2 + (4*n + 6)*x^3)*deriv(v[i - 1], x))/(n + n^2*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-b13531ad79e7c629","steps":["basis A013609","\u03b8+1"],"edges":["e-4579edc5f7a4d216","e-b2171f799212f32b"],"text":"A008459 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A008459 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}
