{"id":"n-834ebd8a3ebc5b93","revision":0,"first_index":0,"fingerprint":"f6-8aaabef611692c95","name":"theta_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","6"],["3","12"],["5","30","27"],["8","60","81"],["13","120","243","108"],["21","228","594","432"],["34","426","1377","1512","405"],["55","780","2997","4320","2025"],["89","1410","6291","11340","8100","1458"],["144","2520","12798","27648","26325","8748"],["233","4464","25434","64152","76950","39366","5103"],["377","7848","49572","142992","206955","142884","35721"],["610","13710","95094","308880","524475","459270","178605","17496"],["987","23820","179982","650160","1267650","1347192","714420","139968"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","8","15","62","149","484","1275","3754","10177","28688","78183","215702","586349","1598140","4324179"],"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":[["30","108","54"],["30","153","135"],["0","45","108"],["0","0","27"]]},"terms":[{"coeff":{"coeffs":[["30","108","54","0"],["90","684","1539","972"],["0","180","810","810"],["0","0","81","162"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-25","-315","-1422","-2754","-1944"],["0","-75","-630","-1755","-1620"],["0","0","-45","-243","-324"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173872,"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-06809d2b4f354736"]},{"id":173873,"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-06809d2b4f354736"]},{"id":173874,"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-06809d2b4f354736"]}],"region_exact":{},"incoming_edges":[{"id":"e-06809d2b4f354736","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-834ebd8a3ebc5b93","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":"430bfffe2371233890107942334df5a969dc61bd427389b2a74a70a6e8eed071","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-b6e50241110d60d5","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","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":[{"id":"l-4a7e4e2cf640b592","kind":"fingerprint_related","node_a_id":"n-16710df6d06121b8","node_b_id":"n-834ebd8a3ebc5b93","fingerprint":"f6-8aaabef611692c95","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8aaabef611692c95"},"other_id":"n-16710df6d06121b8","other":{"id":"n-16710df6d06121b8","name":"theta_plus_one of basis_map_A013610 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","first_index":0,"fingerprint":"f6-8aaabef611692c95","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","6"]],"card_rows":[["1"],["1"],["2","6"],["3","12"]],"row_sums_preview":["1","1","8","15","62","149","484","1275"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["basis A013610","\u03b8+1"],"edges":["e-397aac32dd725997","e-6e7360c5ea4611fd"],"text":"A011973 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A011973 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-550ec7597fe4464b","kind":"fingerprint_related","node_a_id":"n-045e1c27e91fdcc6","node_b_id":"n-834ebd8a3ebc5b93","fingerprint":"f6-8aaabef611692c95","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8aaabef611692c95"},"other_id":"n-045e1c27e91fdcc6","other":{"id":"n-045e1c27e91fdcc6","name":"theta_plus_one of basis_map_A013610 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-8aaabef611692c95","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","6"]],"card_rows":[["1"],["1"],["2","6"],["3","12"]],"row_sums_preview":["1","1","8","15","62","149","484","1275"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["basis A013610","\u03b8+1"],"edges":["e-681c570ae869844f","e-152ea549bd52cc2f"],"text":"A052553 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A052553 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}}],"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,    6;\n  2:  3,   12;\n  3:  5,   30,   27;\n  4:  8,   60,   81;\n  5: 13,  120,  243,   108;\n  6: 21,  228,  594,   432;\n  7: 34,  426, 1377,  1512,  405;\n  8: 55,  780, 2997,  4320, 2025;\n  9: 89, 1410, 6291, 11340, 8100, 1458;\n","recurrence_sentence":"The row polynomials satisfy (30*n+30+(45*n^2+153*n+108)*x+(27*n^3+108*n^2+135*n+54)*x^2)*P(n,x) = (90*n+30+(180*n^2+684*n+108)*x+(81*n^3+810*n^2+1539*n+54)*x^2+(162*n^3+810*n^2+972*n)*x^3)*P(n-1,x) - (25+(75*n+315)*x+(45*n^2+630*n+1422)*x^2+(243*n^2+1755*n+2754)*x^3+(324*n^2+1620*n+1944)*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[((90*n + 30 + (180*n^2 + 684*n + 108)*x + (81*n^3 + 810*n^2 + 1539*n + 54)*x^2 + (162*n^3 + 810*n^2 + 972*n)*x^3)*P[n - 1, x] - (25 + (75*n + 315)*x + (45*n^2 + 630*n + 1422)*x^2 + (243*n^2 + 1755*n + 2754)*x^3 + (324*n^2 + 1620*n + 1944)*x^4)*D[P[n - 1, x], x])/(30*n + 30 + (45*n^2 + 153*n + 108)*x + (27*n^3 + 108*n^2 + 135*n + 54)*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] = ((90*n + 30 + (180*n^2 + 684*n + 108)*x + (81*n^3 + 810*n^2 + 1539*n + 54)*x^2 + (162*n^3 + 810*n^2 + 972*n)*x^3)*v[i - 1] - (25 + (75*n + 315)*x + (45*n^2 + 630*n + 1422)*x^2 + (243*n^2 + 1755*n + 2754)*x^3 + (324*n^2 + 1620*n + 1944)*x^4)*deriv(v[i - 1], x))/(30*n + 30 + (45*n^2 + 153*n + 108)*x + (27*n^3 + 108*n^2 + 135*n + 54)*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-569a801b809b6abd","steps":["basis A013610","\u03b8+1"],"edges":["e-62f41b682ce1c6f7","e-06809d2b4f354736"],"text":"A169803 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A169803 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}