{"id":"n-1f4d09521acf5bb3","revision":1,"first_index":0,"fingerprint":"f6-8c59810c7b2f08c6","name":"theta_plus_one_squared of Riordan array (1+2x,x(1+x)).","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"],["0","12","9"],["0","8","36","16"],["0","0","45","80","25"],["0","0","18","144","150","36"],["0","0","0","112","350","252","49"],["0","0","0","32","400","720","392","64"],["0","0","0","0","225","1080","1323","576","81"],["0","0","0","0","50","900","2450","2240","810","100"],["0","0","0","0","0","396","2695","4928","3564","1100","121"],["0","0","0","0","0","72","1764","6720","9072","5400","1452","144"],["0","0","0","0","0","0","637","5824","14742","15600","7865","1872","169"],["0","0","0","0","0","0","98","3136","15876","29400","25410","11088","2366","196"],["0","0","0","0","0","0","0","960","11340","37800","54450","39600","15210","2940","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","21","60","150","348","763","1608","3285","6550","12804","24624","46709","87570","162525"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["0","0"],["1","-1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","-2","-2"],["4","3","-2"],["2","5","0"],["-2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","-1"],["0","4","-3","-1"],["0","4","1","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160932,"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-d87fb5588738167f"]},{"id":160933,"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-d87fb5588738167f"]},{"id":160934,"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-d87fb5588738167f"]}],"region_exact":{},"incoming_edges":[{"id":"e-d87fb5588738167f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-1f4d09521acf5bb3","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":"06f48c2baf850a025ee8aa35ab771568db960cf6c070311dfcc2ed8367f64433","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-c27b11147fce8bfa","role":"sequence","name":"Riordan array (1+2x,x(1+x)).","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-1188e939c5c82fb7","kind":"fingerprint_related","node_a_id":"n-c8ff64deba2b550e","node_b_id":"n-1f4d09521acf5bb3","fingerprint":"f6-8c59810c7b2f08c6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8c59810c7b2f08c6"},"other_id":"n-c8ff64deba2b550e","other":{"id":"n-c8ff64deba2b550e","name":"derivative of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-8c59810c7b2f08c6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1","2"]],"card_rows":[["1"],["1","2"],["0","4","3"],["0","2","9","4"]],"row_sums_preview":["0","1","3","7","15","30","58","109"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-392f49b496f79fbf","steps":["D"],"edges":["e-1b8904cecd607ff6"],"text":"A026729 \u2192 D"},"headline":"A026729 \u2192 D","chain":null}},{"id":"l-21b0b479e183693b","kind":"fingerprint_related","node_a_id":"n-837b594f6f98209d","node_b_id":"n-1f4d09521acf5bb3","fingerprint":"f6-8c59810c7b2f08c6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8c59810c7b2f08c6"},"other_id":"n-837b594f6f98209d","other":{"id":"n-837b594f6f98209d","name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).","first_index":0,"fingerprint":"f6-8c59810c7b2f08c6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["0","4","3"]],"card_rows":[["1"],["1","2"],["0","4","3"],["0","2","9","4"]],"row_sums_preview":["1","3","7","15","30","58","109","201"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-21854f4257effb01","steps":["\u03b8+1"],"edges":["e-51286c73fb325206"],"text":"A030528 \u2192 \u03b8+1"},"headline":"A030528 \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,  4;\n  2: 0, 12,  9;\n  3: 0,  8, 36,  16;\n  4: 0,  0, 45,  80,  25;\n  5: 0,  0, 18, 144, 150,   36;\n  6: 0,  0,  0, 112, 350,  252,   49;\n  7: 0,  0,  0,  32, 400,  720,  392,   64;\n  8: 0,  0,  0,   0, 225, 1080, 1323,  576,  81;\n  9: 0,  0,  0,   0,  50,  900, 2450, 2240, 810, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^3+n^2-n^2*x)*P(n,x) = -(2*n^3-2*n^2-4*n-(5*n^2+3*n-2)*x+(2*n+2)*x^2)*P(n-1,x) + ((4*n^2+4*n)*x+(n^2-3*n-4)*x^2-(n+1)*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[(-(2*n^3 - 2*n^2 - 4*n - (5*n^2 + 3*n - 2)*x + (2*n + 2)*x^2)*P[n - 1, x] + ((4*n^2 + 4*n)*x + (n^2 - 3*n - 4)*x^2 - (n + 1)*x^3)*D[P[n - 1, x], x])/(n^3 + n^2 - 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] = (-(2*n^3 - 2*n^2 - 4*n - (5*n^2 + 3*n - 2)*x + (2*n + 2)*x^2)*v[i - 1] + ((4*n^2 + 4*n)*x + (n^2 - 3*n - 4)*x^2 - (n + 1)*x^3)*deriv(v[i - 1], x))/(n^3 + n^2 - 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-c27b11147fce8bfa","steps":["(\u03b8+1)\u00b2"],"edges":["e-d87fb5588738167f"],"text":"A113214 \u2192 (\u03b8+1)\u00b2"},"headline":"A113214 \u2192 (\u03b8+1)\u00b2","chain":null}