{"id":"n-83c12a3b577fa369","revision":0,"first_index":0,"fingerprint":"f6-6e5b1fb41454792a","name":"theta_plus_one of veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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":[["4"],["8","2"],["12","12"],["16","38","3"],["20","88","24"],["24","170","102","4"],["28","292","312","40"],["32","462","777","212","5"],["36","688","1680","800","60"],["40","978","3276","2424","380","6"],["44","1340","5904","6288","1700","84"],["48","1782","9999","14520","6050","618","7"],["52","2312","16104","30624","18260","3192","112"],["56","2938","24882","60060","48620","13026","938","8"],["60","3668","37128","110968","117260","44772","5488","144"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["4","10","24","57","132","300","672","1488","3264","7104","15360","33024","70656","150528","319488"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4"]],"recurrence":{"denominator":{"coeffs":[["0","48","-12","6"],["-32","8","-32","11"],["0","16","-28","6"],["0","16","-8","1"]]},"terms":[{"coeff":{"coeffs":[["-32","152","-24","24","0"],["-32","-40","-88","10","6"],["0","32","-12","-19","5"],["0","16","8","-7","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-64","112","-104","68","-12"],["0","0","96","-160","74","-10"],["0","0","32","-48","18","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173076,"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-86adbc40ba138cdf"]},{"id":173077,"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-86adbc40ba138cdf"]},{"id":173078,"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-86adbc40ba138cdf"]}],"region_exact":{},"incoming_edges":[{"id":"e-86adbc40ba138cdf","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-83c12a3b577fa369","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":"b173687c66212bad5e27e29e7446e8f47ff73980752c50edf6f65cc3634fdbdc","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-6feee566fc77ea06","role":"sequence","name":"veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:  4;\n  1:  8,   2;\n  2: 12,  12;\n  3: 16,  38,    3;\n  4: 20,  88,   24;\n  5: 24, 170,  102,    4;\n  6: 28, 292,  312,   40;\n  7: 32, 462,  777,  212,   5;\n  8: 36, 688, 1680,  800,  60;\n  9: 40, 978, 3276, 2424, 380, 6;\n","recurrence_sentence":"The row polynomials satisfy -(32*n-(16*n^3+16*n^2+8*n+48)*x+(8*n^3+28*n^2+32*n+12)*x^2-(n^3+6*n^2+11*n+6)*x^3)*P(n,x) = -(32*n+32-(16*n^3+32*n^2-40*n+152)*x-(8*n^3-12*n^2-88*n-24)*x^2+(7*n^3+19*n^2-10*n-24)*x^3-(n^3+5*n^2+6*n)*x^4)*P(n-1,x) - (64*x-(32*n^2+96*n+112)*x^2+(48*n^2+160*n+104)*x^3-(18*n^2+74*n+68)*x^4+(2*n^2+10*n+12)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4.","mathematica":"P[0, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(32*n + 32 - (16*n^3 + 32*n^2 - 40*n + 152)*x - (8*n^3 - 12*n^2 - 88*n - 24)*x^2 + (7*n^3 + 19*n^2 - 10*n - 24)*x^3 - (n^3 + 5*n^2 + 6*n)*x^4)*P[n - 1, x] - (64*x - (32*n^2 + 96*n + 112)*x^2 + (48*n^2 + 160*n + 104)*x^3 - (18*n^2 + 74*n + 68)*x^4 + (2*n^2 + 10*n + 12)*x^5)*D[P[n - 1, x], x])/(-32*n + (16*n^3 + 16*n^2 + 8*n + 48)*x - (8*n^3 + 28*n^2 + 32*n + 12)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4; for(n = 1, N, my(i = n + 1); v[i] = (-(32*n + 32 - (16*n^3 + 32*n^2 - 40*n + 152)*x - (8*n^3 - 12*n^2 - 88*n - 24)*x^2 + (7*n^3 + 19*n^2 - 10*n - 24)*x^3 - (n^3 + 5*n^2 + 6*n)*x^4)*v[i - 1] - (64*x - (32*n^2 + 96*n + 112)*x^2 + (48*n^2 + 160*n + 104)*x^3 - (18*n^2 + 74*n + 68)*x^4 + (2*n^2 + 10*n + 12)*x^5)*deriv(v[i - 1], x))/(-32*n + (16*n^3 + 16*n^2 + 8*n + 48)*x - (8*n^3 + 28*n^2 + 32*n + 12)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^3)); 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-f48bed27dbc29c5f","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-a61d9f5d8a4277bd","e-86adbc40ba138cdf"],"text":"A391533 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A391533 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}