{"id":"n-5690eebc47d1b0fb","revision":0,"first_index":0,"fingerprint":"f6-985010db292e3b0e","name":"theta_plus_one of reversal of Number triangle T(n,k)=C(n,k)C(2,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","4"],["1","8","3"],["1","12","9"],["1","16","18"],["1","20","30"],["1","24","45"],["1","28","63"],["1","32","84"],["1","36","108"],["1","40","135"],["1","44","165"],["1","48","198"],["1","52","234"],["1","56","273"]],"degrees":[0,1,2,2,2,2,2,2,2,2,2,2,2,2,2],"row_sums":["1","5","12","22","35","51","70","92","117","145","176","210","247","287","330"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"],["1","8","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171275,"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-6cdb196bf473a665"]},{"id":171276,"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-6cdb196bf473a665"]},{"id":171277,"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-6cdb196bf473a665"]}],"region_exact":{},"incoming_edges":[{"id":"e-6cdb196bf473a665","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-5690eebc47d1b0fb","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":"035c826ea234fad6c4b55b84220f78eca6f77881c55751db471a69c4187bb660","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-88b9d8361c4b11db","role":"sequence","name":"reversal of Number triangle T(n,k)=C(n,k)C(2,n-k).","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: 1,  4;\n  2: 1,  8,   3;\n  3: 1, 12,   9;\n  4: 1, 16,  18;\n  5: 1, 20,  30;\n  6: 1, 24,  45;\n  7: 1, 28,  63;\n  8: 1, 32,  84;\n  9: 1, 36, 108;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-3,x) - 3*P(n-2,x) + 3*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 8*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 8*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[P[n - 3, x] - 3*P[n - 2, x] + 3*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 8*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = v[i - 3] - 3*v[i - 2] + 3*v[i - 1]); 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-1815d3e1c9c57d5b","steps":["reverse","\u03b8+1"],"edges":["e-29732eeeac05382b","e-6cdb196bf473a665"],"text":"A106246 \u2192 reverse \u2192 \u03b8+1"},"headline":"A106246 \u2192 reverse \u2192 \u03b8+1","chain":null}