{"id":"n-a99b6ab79c8d82f3","revision":0,"first_index":1,"fingerprint":"f6-525575d96ecebcc3","name":"mobius_x_over_x_plus_one of reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*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"],["1","2"],["1","5","5"],["1","8","19","13"],["1","11","42","65","34"],["1","14","74","183","210","89"],["1","17","115","394","717","654","233"],["1","20","165","725","1825","2622","1985","610"],["1","23","224","1203","3885","7703","9134","5911","1597"],["1","26","292","1855","7329","18633","30418","30691","17345","4181"],["1","29","369","2708","12670","39417","82432","114242","100284","50305","10946"],["1","32","455","3789","20502","75572","193354","342905","412592","320466","144516","28657"],["1","35","550","5125","31500","134370","407400","883550","1358875","1444000","1005630","411900","75025"],["1","38","654","6743","46420","225081","790008","2030178","3816171","5177720","4925038","3108324","1166209","196418"],["1","41","767","8670","66099","359216","1433751","4265832","9499305","15742683","19099323","16439438","9485551","3283145","514229"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","41","153","571","2131","7953","29681","110771","413403","1542841","5757961","21489003","80198051"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-3","-6","-3"],["2","4","2"]]},"terms":[{"coeff":{"coeffs":[["-3","-22","-45","-36","-10"],["2","13","25","19","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-8","-18","-16","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172498,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-54063cacb4a629fa"]},{"id":172499,"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-54063cacb4a629fa"]},{"id":172500,"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-54063cacb4a629fa"]}],"region_exact":{},"incoming_edges":[{"id":"e-54063cacb4a629fa","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-a99b6ab79c8d82f3","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"6a5738a87f6740ad3952f856e9f09f5440057b93dda51ae55d7ca0900c330c40","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d8cfb1e39c5d56b0","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*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-1ec20e8497146b58","kind":"fingerprint_related","node_a_id":"n-acbcb73ba2185f71","node_b_id":"n-a99b6ab79c8d82f3","fingerprint":"f6-525575d96ecebcc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-525575d96ecebcc3"},"other_id":"n-acbcb73ba2185f71","other":{"id":"n-acbcb73ba2185f71","name":"mobius_x_over_x_plus_one of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-525575d96ecebcc3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","8","19","13"]],"row_sums_preview":["1","3","11","41","153","571","2131","7953"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6758fe7729845be8","steps":["x \u21a6 x/(x+1)"],"edges":["e-2bcd3444a1b3af35"],"text":"A054142 \u2192 x \u21a6 x/(x+1)"},"headline":"A054142 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-48f0117799f6bf72","kind":"fingerprint_related","node_a_id":"n-e33a28c556ae59b3","node_b_id":"n-a99b6ab79c8d82f3","fingerprint":"f6-525575d96ecebcc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-525575d96ecebcc3"},"other_id":"n-e33a28c556ae59b3","other":{"id":"n-e33a28c556ae59b3","name":"mobius_x_over_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-525575d96ecebcc3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","1","5","5"]],"row_sums_preview":["1","1","3","11","41","153","571","2131"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3af57aaf5c129e4b","steps":["x \u21a6 x/(x+1)"],"edges":["e-89399084540755ad"],"text":"A121314 \u2192 x \u21a6 x/(x+1)"},"headline":"A121314 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7acc29058d98f422","kind":"fingerprint_related","node_a_id":"n-d0986a7acf60271a","node_b_id":"n-a99b6ab79c8d82f3","fingerprint":"f6-525575d96ecebcc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-525575d96ecebcc3"},"other_id":"n-d0986a7acf60271a","other":{"id":"n-d0986a7acf60271a","name":"shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-525575d96ecebcc3","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["5","5","1"]],"row_sums_preview":["1","1","3","11","41","153","571","2131"],"regions":["(-5, -1)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01fe968f3e05a927","steps":["x \u21a6 x+1"],"edges":["e-17055ffc05dec6ce"],"text":"A165253 \u2192 x \u21a6 x+1"},"headline":"A165253 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-7eb1b6d35581324b","kind":"fingerprint_related","node_a_id":"n-a99b6ab79c8d82f3","node_b_id":"n-cc6686414670682e","fingerprint":"f6-525575d96ecebcc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-525575d96ecebcc3"},"other_id":"n-cc6686414670682e","other":{"id":"n-cc6686414670682e","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-525575d96ecebcc3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","8"],["16","80","80"]],"card_rows":[["1"],["4","8"],["16","80","80"],["64","512","1216","832"]],"row_sums_preview":["1","12","176","2624","39168","584704","8728576","130301952"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-36be6b9e5c42de00","e-dc0ff6195ac36166"],"text":"A103327 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A103327 \u2192 basis A013611 \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: 1,  2;\n   3: 1,  5,   5;\n   4: 1,  8,  19,   13;\n   5: 1, 11,  42,   65,   34;\n   6: 1, 14,  74,  183,  210,    89;\n   7: 1, 17, 115,  394,  717,   654,   233;\n   8: 1, 20, 165,  725, 1825,  2622,  1985,   610;\n   9: 1, 23, 224, 1203, 3885,  7703,  9134,  5911,  1597;\n  10: 1, 26, 292, 1855, 7329, 18633, 30418, 30691, 17345, 4181;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (2*n-3+(13*n-22)*x+(25*n-45)*x^2+(19*n-36)*x^3+(5*n-10)*x^4)*P(n-1,x) - (x+8*x^2+18*x^3+16*x^4+5*x^5)*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[((2*n - 3 + (13*n - 22)*x + (25*n - 45)*x^2 + (19*n - 36)*x^3 + (5*n - 10)*x^4)*P[n - 1, x] - (x + 8*x^2 + 18*x^3 + 16*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)]];\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] = ((2*n - 3 + (13*n - 22)*x + (25*n - 45)*x^2 + (19*n - 36)*x^3 + (5*n - 10)*x^4)*v[i - 1] - (x + 8*x^2 + 18*x^3 + 16*x^4 + 5*x^5)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^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-0eba401b117f16a9","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-6116c33129f68e6c","e-54063cacb4a629fa"],"text":"A085478 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A085478 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}