{"id":"n-6a5cb0b957c123da","revision":0,"first_index":1,"fingerprint":"f6-73a02a3902b59fb9","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","2"],["28","24","4"],["208","264","96","8"],["1552","2624","1504","320","16"],["11584","24480","19200","6720","960","32"],["86464","219264","218304","106752","25920","2688","64"],["645376","1909376","2304512","1448832","499968","90496","7168","128"],["4817152","16287744","23100416","17758208","7964160","2078720","293888","18432","256"],["35955712","136770048","222879744","202598400","111937536","38329344","7913472","903168","46080","512"],["268377088","1134295040","2088145920","2191319040","1439569920","611323904","166963200","28139520","2657280","112640","1024"],["2003193856","9313142784","19112845312","22742624256","17321779200","8789659648","2994962432","672731136","94752768","7546880","270336","2048"],["14952042496","75833720832","171654881280","228384751616","197903314944","116877262848","47801614336","13468139520","2545348608","305119232","20815872","638976","4096"],["111603564544","613199814656","1517604634624","2232571707392","2169086033920","1461927993344","698337722368","237142310912","56494637056","9143214080","946536448","56016896","1490944","8192"],["833020346368","4929061552128","13240538873856","21340879257600","22977664303104","17407493505024","9516575932416","3786694852608","1091393372160","223668436992","31439536128","2844721152","147603456","3440640","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","56","576","6016","62976","659456","6905856","72318976","757334016","7930904576","83053510656","869747654656","9108115685376","95381425750016"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-20","-12"],["10","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-10","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-2729d81eb6e0aef8","recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","-8","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182235,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-2fbe57238715d255"]},{"id":182236,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-2fbe57238715d255","e-5f244728fd5e42e0"]},{"id":182237,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-2fbe57238715d255","e-5f244728fd5e42e0"]},{"id":182238,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-2fbe57238715d255","e-5f244728fd5e42e0"]},{"id":182239,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-5f244728fd5e42e0"]}],"region_exact":{},"incoming_edges":[{"id":"e-5f244728fd5e42e0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6a5cb0b957c123da","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a44140acd6290b3416913a8fa38c130adf1432080823d73b9cc32c66c17bedea","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-91f141c4f7274be3","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-2729d81eb6e0aef8","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-6a5cb0b957c123da","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"0f5e5773d5eada0303babc1276088814d0cc87e47492af84de4f9ee970d75c85","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-2fbe57238715d255","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-6a5cb0b957c123da","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"802d60a391c1dfe44d7c7b824edfef67a2b6efc0c0615b30608c47747a976737","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-91f141c4f7274be3","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-8d0c9c7cc9060358","kind":"fingerprint_related","node_a_id":"n-e9f3812507785414","node_b_id":"n-6a5cb0b957c123da","fingerprint":"f6-73a02a3902b59fb9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73a02a3902b59fb9"},"other_id":"n-e9f3812507785414","other":{"id":"n-e9f3812507785414","name":"mobius_x_over_x_plus_one of shift_x_plus_one of chebyshev_t","first_index":0,"fingerprint":"f6-73a02a3902b59fb9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","7"]],"card_rows":[["1"],["1","2"],["1","6","7"],["1","12","33","26"]],"row_sums_preview":["1","3","14","72","376","1968","10304","53952"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9ae8d431f7b33f22","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-298ee5f2391f30e9","e-3d5368d261b111d1"],"text":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        4,         2;\n   3:       28,        24,         4;\n   4:      208,       264,        96,         8;\n   5:     1552,      2624,      1504,       320,        16;\n   6:    11584,     24480,     19200,      6720,       960,       32;\n   7:    86464,    219264,    218304,    106752,     25920,     2688,      64;\n   8:   645376,   1909376,   2304512,   1448832,    499968,    90496,    7168,    128;\n   9:  4817152,  16287744,  23100416,  17758208,   7964160,  2078720,  293888,  18432,   256;\n  10: 35955712, 136770048, 222879744, 202598400, 111937536, 38329344, 7913472, 903168, 46080, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (10*n-20+(6*n-12)*x)*P(n-1,x) - (6+10*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((10*n - 20 + (6*n - 12)*x)*P[n - 1, x] - (6 + 10*x + 4*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = ((10*n - 20 + (6*n - 12)*x)*v[i - 1] - (6 + 10*x + 4*x^2)*deriv(v[i - 1], x))/(n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-55a1228129c4451e","e-2fbe57238715d255"],"text":"A086645 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A086645 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}