{"id":"n-3f9e71a819c8f308","revision":1,"first_index":0,"fingerprint":"f6-8de4b75c4559d58b","name":"theta_plus_one 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","2"],["0","6","3"],["0","4","12","4"],["0","0","15","20","5"],["0","0","6","36","30","6"],["0","0","0","28","70","42","7"],["0","0","0","8","80","120","56","8"],["0","0","0","0","45","180","189","72","9"],["0","0","0","0","10","150","350","280","90","10"],["0","0","0","0","0","66","385","616","396","110","11"],["0","0","0","0","0","12","252","840","1008","540","132","12"],["0","0","0","0","0","0","91","728","1638","1560","715","156","13"],["0","0","0","0","0","0","14","392","1764","2940","2310","924","182","14"],["0","0","0","0","0","0","0","120","1260","3780","4950","3300","1170","210","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","9","20","40","78","147","272","495","890","1584","2796","4901","8540","14805"],"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":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","2"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160929,"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-8fa65daf09475d6c"]},{"id":160930,"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-8fa65daf09475d6c"]},{"id":160931,"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-8fa65daf09475d6c"]}],"region_exact":{},"incoming_edges":[{"id":"e-8fa65daf09475d6c","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-3f9e71a819c8f308","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":"34a97956f341195b8b7d2fcac3d449d42e5e55fc3c212b003425e3513f9e9086","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-77cd9a8871834fd8","kind":"fingerprint_related","node_a_id":"n-392f49b496f79fbf","node_b_id":"n-3f9e71a819c8f308","fingerprint":"f6-8de4b75c4559d58b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8de4b75c4559d58b"},"other_id":"n-392f49b496f79fbf","other":{"id":"n-392f49b496f79fbf","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-8de4b75c4559d58b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","1"]],"card_rows":[["1"],["0","1"],["0","1","1"],["0","0","2","1"]],"row_sums_preview":["1","1","2","3","5","8","13","21"],"regions":["(-inf, 0]","[-4, 0]","(-4, 0]"],"region_exact":{},"oeis_ids":["A026729"],"derivation":{"root":"n-392f49b496f79fbf","steps":[],"edges":[],"text":"A026729"},"headline":"A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","chain":null}},{"id":"l-82ff4deed20f32f4","kind":"fingerprint_related","node_a_id":"n-c8b3f39d03de798f","node_b_id":"n-3f9e71a819c8f308","fingerprint":"f6-8de4b75c4559d58b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8de4b75c4559d58b"},"other_id":"n-c8b3f39d03de798f","other":{"id":"n-c8b3f39d03de798f","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-8de4b75c4559d58b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","2"]],"row_sums_preview":["1","1","2","3","5","8","13","21"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A052553"],"derivation":{"root":"n-c8b3f39d03de798f","steps":[],"edges":[],"text":"A052553"},"headline":"A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","chain":null}},{"id":"l-a258cc7f3b992eef","kind":"fingerprint_related","node_a_id":"n-75ccd0453a9b6cec","node_b_id":"n-3f9e71a819c8f308","fingerprint":"f6-8de4b75c4559d58b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8de4b75c4559d58b"},"other_id":"n-75ccd0453a9b6cec","other":{"id":"n-75ccd0453a9b6cec","name":"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":"f6-8de4b75c4559d58b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","2"]],"row_sums_preview":["1","1","2","3","5","8","13","21"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A011973"],"derivation":{"root":"n-75ccd0453a9b6cec","steps":[],"edges":[],"text":"A011973"},"headline":"A011973: 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.","chain":null}},{"id":"l-bde883418717dd34","kind":"fingerprint_related","node_a_id":"n-21854f4257effb01","node_b_id":"n-3f9e71a819c8f308","fingerprint":"f6-8de4b75c4559d58b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8de4b75c4559d58b"},"other_id":"n-21854f4257effb01","other":{"id":"n-21854f4257effb01","name":"Triangle read by rows: T(n,k) = binomial(k,n-k).","first_index":0,"fingerprint":"f6-8de4b75c4559d58b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["0","2","1"]],"card_rows":[["1"],["1","1"],["0","2","1"],["0","1","3","1"]],"row_sums_preview":["1","2","3","5","8","13","21","34"],"regions":["(-inf, 0]","[-4, 0]","(-4, 0]"],"region_exact":{},"oeis_ids":["A030528"],"derivation":{"root":"n-21854f4257effb01","steps":[],"edges":[],"text":"A030528"},"headline":"A030528: Triangle read by rows: T(n,k) = binomial(k,n-k).","chain":null}},{"id":"l-d6e885e16ae77ae0","kind":"fingerprint_related","node_a_id":"n-569a801b809b6abd","node_b_id":"n-3f9e71a819c8f308","fingerprint":"f6-8de4b75c4559d58b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8de4b75c4559d58b"},"other_id":"n-569a801b809b6abd","other":{"id":"n-569a801b809b6abd","name":"Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","first_index":0,"fingerprint":"f6-8de4b75c4559d58b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","2"]],"card_rows":[["1"],["1","1"],["1","2"],["1","3","1"]],"row_sums_preview":["1","2","3","5","8","13","21","34"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A169803"],"derivation":{"root":"n-569a801b809b6abd","steps":[],"edges":[],"text":"A169803"},"headline":"A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","chain":null}},{"id":"l-06395fe5631c5a3e","kind":"oeis_match","node_a_id":"n-3f9e71a819c8f308","node_b_id":null,"fingerprint":null,"oeis_id":"A023607","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A023607","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A023607","relation":"match_row_sums"}],"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, 2;\n  2: 0, 6,  3;\n  3: 0, 4, 12,  4;\n  4: 0, 0, 15, 20,  5;\n  5: 0, 0,  6, 36, 30,   6;\n  6: 0, 0,  0, 28, 70,  42,   7;\n  7: 0, 0,  0,  8, 80, 120,  56,   8;\n  8: 0, 0,  0,  0, 45, 180, 189,  72,  9;\n  9: 0, 0,  0,  0, 10, 150, 350, 280, 90, 10;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(2*n-4-2*x)*P(n-1,x) + (4*x+x^2)*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 - 4 - 2*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(n)]];\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 - 4 - 2*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(n)); 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"],"edges":["e-8fa65daf09475d6c"],"text":"A113214 \u2192 \u03b8+1"},"headline":"A113214 \u2192 \u03b8+1","chain":null}