{"id":"n-7b4f74988046113d","revision":1,"first_index":0,"fingerprint":"f6-6832a290ef8b97c2","name":"theta_plus_one of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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"],["1","4"],["1","10","15"],["1","18","63","56"],["1","28","168","336","210"],["1","40","360","1200","1650","792"],["1","54","675","3300","7425","7722","3003"],["1","70","1155","7700","25025","42042","35035","11440"],["1","88","1848","16016","70070","168168","224224","155584","43758"],["1","108","2808","30576","171990","550368","1039584","1145664","680238","167960"],["1","130","4095","54600","382200","1559376","3898440","6046560","5668650","2939300","646646"],["1","154","5775","92400","785400","3958416","12534984","25581600","33575850","27358100","12584726","2496144"],["1","180","7920","149600","1514700","9209376","35814240","92093760","158286150","179781800","129442896","53488800","9657700"],["1","208","10608","233376","2771340","19953648","93117024","292653504","631034118","934865360","934865360","602640480","225990180","37442160"],["1","238","13923","352716","4849845","40738698","224062839","841378824","2208619413","4090035950","5317046735","4745793780","2768379705","950094810","145422675"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","138","743","4043","22180","122468","679757","3789297","21199998","118973550","669447123","3775577367","21336790152"],"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"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","4"],["2","6"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"],["0","2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164919,"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-10c4942a23949b48"]},{"id":164920,"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-10c4942a23949b48"]},{"id":164921,"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-10c4942a23949b48"]},{"id":164922,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-10c4942a23949b48"]}],"region_exact":{},"incoming_edges":[{"id":"e-10c4942a23949b48","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-7b4f74988046113d","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":"c946e27b722c12570a7c2aee47010c130ad9017cd33a5588875cb6dd512127ab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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-112db8be4020d3db","kind":"candidate_equal","node_a_id":"n-fc24cdc1c25a5a29","node_b_id":"n-7b4f74988046113d","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-fc24cdc1c25a5a29","other":{"id":"n-fc24cdc1c25a5a29","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-6832a290ef8b97c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","10","15"]],"card_rows":[["1"],["1","4"],["1","10","15"],["1","18","63","56"]],"row_sums_preview":["1","5","26","138","743","4043","22180","122468"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ca874876022e136","steps":["x \u21a6 x/(x+1)"],"edges":["e-f6a7bd41834fc468"],"text":"A062196 \u2192 x \u21a6 x/(x+1)"},"headline":"A062196 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-42b955ccbadd2d36","kind":"oeis_match","node_a_id":"n-7b4f74988046113d","node_b_id":null,"fingerprint":null,"oeis_id":"A035029","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A035029","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A035029","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-7b4f74988046113d","oeis_id":"A035029","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":16081}],"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,  10,   15;\n  3: 1,  18,   63,    56;\n  4: 1,  28,  168,   336,    210;\n  5: 1,  40,  360,  1200,   1650,    792;\n  6: 1,  54,  675,  3300,   7425,   7722,    3003;\n  7: 1,  70, 1155,  7700,  25025,  42042,   35035,   11440;\n  8: 1,  88, 1848, 16016,  70070, 168168,  224224,  155584,  43758;\n  9: 1, 108, 2808, 30576, 171990, 550368, 1039584, 1145664, 680238, 167960;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n)*P(n,x) = (n^2+2*n+(2*n^2+6*n+4)*x)*P(n-1,x) + ((2*n+2)*x+(2*n+2)*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[((n^2 + 2*n + (2*n^2 + 6*n + 4)*x)*P[n - 1, x] + ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*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] = ((n^2 + 2*n + (2*n^2 + 6*n + 4)*x)*v[i - 1] + ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*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-66e2c3448245b0b7","steps":["\u03b8+1"],"edges":["e-10c4942a23949b48"],"text":"A033282 \u2192 \u03b8+1"},"headline":"A033282 \u2192 \u03b8+1","chain":null}