{"id":"n-24b9ca69c1604c2e","revision":0,"first_index":0,"fingerprint":"f6-a648d2a657733b6a","name":"theta_plus_one of veronese_section_2_0 of Triangle of coefficients in expansion of (1+2*x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","8"],["1","24"],["1","48","48"],["1","80","240"],["1","120","720","256"],["1","168","1680","1792"],["1","224","3360","7168","1280"],["1","288","6048","21504","11520"],["1","360","10080","53760","57600","6144"],["1","440","15840","118272","211200","67584"],["1","528","23760","236544","633600","405504","28672"],["1","624","34320","439296","1647360","1757184","372736"],["1","728","48048","768768","3843840","6150144","2609152","131072"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","9","25","97","321","1097","3641","12033","39361","127945","413337","1328609","4251521","13551753"],"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"]],"recurrence":{"denominator":{"coeffs":[["3","0"],["-3","-2"],["0","0"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["3","24","-16"],["-3","-26","-8"],["0","0","16"],["0","2","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-6","32","-32"],["0","0","12","-48"],["0","0","4","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173900,"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-ec4872e119453564"]},{"id":173901,"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-ec4872e119453564"]},{"id":173902,"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-ec4872e119453564"]}],"region_exact":{},"incoming_edges":[{"id":"e-ec4872e119453564","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-24b9ca69c1604c2e","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":"791ad0f60be6432f607354876abb3af34a86910e7e9e19071f6cb06648e48203","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-e67a7f2789421c3e","role":"sequence","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (1+2*x)^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-2c8fcd1a4ba4dca7","kind":"fingerprint_related","node_a_id":"n-a41cc546cebc5a10","node_b_id":"n-24b9ca69c1604c2e","fingerprint":"f6-a648d2a657733b6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a648d2a657733b6a"},"other_id":"n-a41cc546cebc5a10","other":{"id":"n-a41cc546cebc5a10","name":"theta_plus_one of gamma_polynomial of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-a648d2a657733b6a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","8"]],"card_rows":[["1"],["1"],["1","8"],["1","24"]],"row_sums_preview":["1","1","9","25","97","321","1097","3641"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["\u03b3","\u03b8+1"],"edges":["e-94f70be4e24ec504","e-d78dd07591e13668"],"text":"A086645 \u2192 \u03b3 \u2192 \u03b8+1"},"headline":"A086645 \u2192 \u03b3 \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: 1;\n  2: 1,   8;\n  3: 1,  24;\n  4: 1,  48,   48;\n  5: 1,  80,  240;\n  6: 1, 120,  720,   256;\n  7: 1, 168, 1680,  1792;\n  8: 1, 224, 3360,  7168,  1280;\n  9: 1, 288, 6048, 21504, 11520;\n","recurrence_sentence":"The row polynomials satisfy -(3*n-3-(2*n^3-2*n)*x)*P(n,x) = -(3*n-3-(2*n^3-26*n+24)*x-(8*n^3+16*n^2-8*n-16)*x^2)*P(n-1,x) - (6*x-(4*n^2+12*n+32)*x^2+(16*n^2+48*n+32)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 - (2*n^3 - 26*n + 24)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*P[n - 1, x] - (6*x - (4*n^2 + 12*n + 32)*x^2 + (16*n^2 + 48*n + 32)*x^3)*D[P[n - 1, x], x])/(-3*n + 3 + (2*n^3 - 2*n)*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; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 3 - (2*n^3 - 26*n + 24)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*v[i - 1] - (6*x - (4*n^2 + 12*n + 32)*x^2 + (16*n^2 + 48*n + 32)*x^3)*deriv(v[i - 1], x))/(-3*n + 3 + (2*n^3 - 2*n)*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-71965e79e47a4526","steps":["Veronese 2","\u03b8+1"],"edges":["e-eb7af842d4a02729","e-ec4872e119453564"],"text":"A013609 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A013609 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}
