{"id":"n-790e61cd1a4f1614","revision":0,"first_index":0,"fingerprint":"f6-d88a651a3889a86b","name":"theta_plus_one of reversal of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4","12","3"],["8","40","36","4"],["16","112","180","80","5"],["32","288","672","560","150","6"],["64","704","2160","2688","1400","252","7"],["128","1664","6336","10560","8400","3024","392","8"],["256","3840","17472","36608","39600","22176","5880","576","9"],["512","8704","46080","116480","160160","123552","51744","10560","810","10"],["1024","19456","117504","348160","582400","576576","336336","109824","17820","1100","11"],["2048","43008","291840","992256","1958400","2376192","1793792","823680","216216","28600","1452","12"],["4096","94208","709632","2723840","6201600","8930304","8316672","4978688","1853280","400400","44044","1872","13"],["8192","204800","1695744","7254016","18726400","31256064","34728960","25798656","12602304","3889600","704704","65520","2366","14"],["16384","442368","3993600","18841600","54405120","103369728","133706496","119070720","72558720","29560960","7701408","1188096","94640","2940","15"],["32768","950272","9289728","47923200","153088000","326430720","482392064","502081536","368375040","188115200","65034112","14511744","1930656","133280","3600","16"],["65536","2031616","21381120","119734272","419328000","992010240","1650288640","1968947200","1694525184","1046003200","455238784","135442944","26201760","3038784","183600","4352","17"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["19","88","393","1708","7275","30512","126417","518612","2110211","8527496","34258649","136937340","544952795","2160291936","8534425249"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","12","3"]],"recurrence":{"denominator":{"coeffs":[["12","18"],["14","33"],["4","20"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["24","92","73"],["28","142","131"],["8","64","72"],["0","8","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-39","-46"],["0","-4","-28","-40"],["0","0","-4","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177615,"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-416643a040539a36"]},{"id":177616,"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-416643a040539a36"]},{"id":177617,"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-416643a040539a36"]}],"region_exact":{},"incoming_edges":[{"id":"e-416643a040539a36","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-790e61cd1a4f1614","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":"1dfe8a8fec9ee521a3523a3158302b002fe016abf05f92367226e010e5c41a57","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-ccea8be4a0f670ab","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=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-48caa9cb62900d2b","kind":"fingerprint_related","node_a_id":"n-790e61cd1a4f1614","node_b_id":"n-ae385526595d7f42","fingerprint":"f6-d88a651a3889a86b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d88a651a3889a86b"},"other_id":"n-ae385526595d7f42","other":{"id":"n-ae385526595d7f42","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-d88a651a3889a86b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["16","48","12"]],"card_rows":[["1"],["4","4"],["16","48","12"],["64","320","288","32"]],"row_sums_preview":["1","8","76","704","6288","54656","465600","3905536"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013609","\u03b8+1"],"edges":["e-49766d0fffe14f49","e-a7b13ca9deb2a4fd"],"text":"A103327 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A103327 \u2192 basis A013609 \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:    4,    12,      3;\n  1:    8,    40,     36,      4;\n  2:   16,   112,    180,     80,       5;\n  3:   32,   288,    672,    560,     150,       6;\n  4:   64,   704,   2160,   2688,    1400,     252,       7;\n  5:  128,  1664,   6336,  10560,    8400,    3024,     392,      8;\n  6:  256,  3840,  17472,  36608,   39600,   22176,    5880,    576,      9;\n  7:  512,  8704,  46080, 116480,  160160,  123552,   51744,  10560,    810,    10;\n  8: 1024, 19456, 117504, 348160,  582400,  576576,  336336, 109824,  17820,  1100,   11;\n  9: 2048, 43008, 291840, 992256, 1958400, 2376192, 1793792, 823680, 216216, 28600, 1452, 12;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+14*n+12+(4*n^3+20*n^2+33*n+18)*x)*P(n,x) = (8*n^2+28*n+24+(8*n^3+64*n^2+142*n+92)*x+(12*n^3+72*n^2+131*n+73)*x^2)*P(n-1,x) - ((4*n+8)*x+(4*n^2+28*n+39)*x^2+(8*n^2+40*n+46)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 12*x + 3*x^2.","mathematica":"P[0, x_] := 4 + 12*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 28*n + 24 + (8*n^3 + 64*n^2 + 142*n + 92)*x + (12*n^3 + 72*n^2 + 131*n + 73)*x^2)*P[n - 1, x] - ((4*n + 8)*x + (4*n^2 + 28*n + 39)*x^2 + (8*n^2 + 40*n + 46)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12 + (4*n^3 + 20*n^2 + 33*n + 18)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 12*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 28*n + 24 + (8*n^3 + 64*n^2 + 142*n + 92)*x + (12*n^3 + 72*n^2 + 131*n + 73)*x^2)*v[i - 1] - ((4*n + 8)*x + (4*n^2 + 28*n + 39)*x^2 + (8*n^2 + 40*n + 46)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12 + (4*n^3 + 20*n^2 + 33*n + 18)*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-8eb9bb3a09f0b7fd","steps":["reverse","\u03b8+1"],"edges":["e-efd270083a78762c","e-416643a040539a36"],"text":"A123519 \u2192 reverse \u2192 \u03b8+1"},"headline":"A123519 \u2192 reverse \u2192 \u03b8+1","chain":null}
