{"id":"n-74c5a3a593f6b651","revision":0,"first_index":1,"fingerprint":"f6-c933affcacfcef22","name":"reversal of basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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"],["2","4"],["4","14","11"],["8","40","62","29"],["16","104","240","230","76"],["32","256","784","1140","780","199"],["64","608","2320","4528","4740","2508","521"],["128","1408","6432","15760","22280","18096","7786","1364"],["256","3200","17024","50208","89520","98504","65120","23578","3571"],["512","7168","43520","150080","323232","449808","403472","224420","70090","9349"],["1024","15872","108288","427520","1079680","1819104","2066848","1560752","748320","205390","24476"],["2048","34816","263680","1172736","3398400","6727168","9267776","8872240","5775424","2431320","595032","64079"],["4096","75776","630784","3120640","10206720","23222016","37637376","43728704","36100720","20628160","7735704","1707864","167761"],["8192","163840","1486848","8098816","29511680","75891456","141542400","193642368","194119776","140686240","71580464","24191520","4864082","439204"],["16384","352256","3461120","20582400","82689024","237197312","500552448","788689920","931529280","820003104","529118928","242496272","74567020","13762790","1149851"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","29","139","666","3191","15289","73254","350981","1681651","8057274","38604719","184966321","886226886","4246168109"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-6"],["4"]]},"terms":[{"coeff":{"coeffs":[["-12","-28","-10"],["8","18","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-12","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177901,"position":0,"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-becb48808f7bde08"]},{"id":177902,"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-becb48808f7bde08"]},{"id":177903,"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-becb48808f7bde08"]}],"region_exact":{},"incoming_edges":[{"id":"e-becb48808f7bde08","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-74c5a3a593f6b651","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"e0151c4ba23088602765b8e729b28a724d0f53cdf979d32af35eecf23a78cca0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-45b8d33479b2056d","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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-f4d0fb21a3011fa1","kind":"fingerprint_related","node_a_id":"n-45b8d33479b2056d","node_b_id":"n-74c5a3a593f6b651","fingerprint":"f6-c933affcacfcef22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c933affcacfcef22"},"other_id":"n-45b8d33479b2056d","other":{"id":"n-45b8d33479b2056d","name":"basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","first_index":1,"fingerprint":"f6-c933affcacfcef22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["11","14","4"]],"card_rows":[["1"],["4","2"],["11","14","4"],["29","62","40","8"]],"row_sums_preview":["1","6","29","139","666","3191","15289","73254"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A013609"],"edges":["e-0d48af5cb3d93d03"],"text":"A111125 \u2192 basis A013609"},"headline":"A111125 \u2192 basis A013609","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:   2,    4;\n   3:   4,   14,    11;\n   4:   8,   40,    62,     29;\n   5:  16,  104,   240,    230,     76;\n   6:  32,  256,   784,   1140,    780,    199;\n   7:  64,  608,  2320,   4528,   4740,   2508,    521;\n   8: 128, 1408,  6432,  15760,  22280,  18096,   7786,   1364;\n   9: 256, 3200, 17024,  50208,  89520,  98504,  65120,  23578,  3571;\n  10: 512, 7168, 43520, 150080, 323232, 449808, 403472, 224420, 70090, 9349;\n","recurrence_sentence":"The row polynomials satisfy (4*n-6)*P(n,x) = (8*n-12+(18*n-28)*x+(5*n-10)*x^2)*P(n-1,x) - (4*x+12*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (18*n - 28)*x + (5*n - 10)*x^2)*P[n - 1, x] - (4*x + 12*x^2 + 5*x^3)*D[P[n - 1, x], x])/(4*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (18*n - 28)*x + (5*n - 10)*x^2)*v[i - 1] - (4*x + 12*x^2 + 5*x^3)*deriv(v[i - 1], x))/(4*n - 6)); 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-318700a428d7b97b","steps":["basis A013609","reverse"],"edges":["e-0d48af5cb3d93d03","e-becb48808f7bde08"],"text":"A111125 \u2192 basis A013609 \u2192 reverse"},"headline":"A111125 \u2192 basis A013609 \u2192 reverse","chain":null}
