{"id":"n-22dc599684e9805b","revision":0,"first_index":0,"fingerprint":"f6-402c4469f5466095","name":"mobius_x_over_x_plus_one of basis_map_A008277 of Cube of lower triangular normalized binomial matrix.","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"],["4","5"],["16","41","26"],["64","253","329","141"],["256","1393","2794","2455","799"],["1024","7221","19919","26976","17989","4736"],["4096","36105","128836","239624","246011","132584","29371"],["16384","176429","784545","1883351","2650864","2196461","994944","190497"],["65536","849377","4592582","13693973","24817591","28139929","19568536","7651985","1291020"],["262144","4050277","26181827","94504754","212067155","308877451","293394499","175868872","60527003","9131275"],["1048576","19202809","146631112","629131062","1702273959","3061137155","3726873921","3046203780","1604829259","493292867","67310847"],["4194304","90771165","811692485","4084461041","13074273838","28253138838","42360781231","44290673252","31766289444","14927317780","4145500611","516369838"],["16777216","428690001","4460932906","26060130667","97288958339","247751108890","445100450796","571843260130","523716953189","334555332572","141865953582","35927920564","4116416797"],["67108864","2026009493","24420820839","164322525940","707642206985","2092507762346","4415946167918","6781520353180","7616903554402","6203455601487","3571364195684","1379510207310","321049570048","34051164985"],["268435456","9593176553","133496452844","1028262284308","5063863574771","17189344601714","41975697341007","75520446457884","101180341596478","100891029135467","73949136982436","38735201051910","13735843436056","2956642510893","291871399682"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","83","787","7697","77865","816627","8893475","100670529","1184865257","14497935347","184325463827","2433132885649","33314787249481","472661038437459"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["4","10","7","1"],["0","-1","-2","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179362,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-a1fa34e479f265c3"]},{"id":179363,"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-a1fa34e479f265c3"]},{"id":179364,"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-a1fa34e479f265c3"]}],"region_exact":{},"incoming_edges":[{"id":"e-a1fa34e479f265c3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-22dc599684e9805b","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":"041882410a167e9ea0539b91e254108ddf92aecb2da5b6d4bdcd447758645b03","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-41442279a7af7481","role":"sequence","name":"basis_map_A008277 of Cube of lower triangular normalized binomial matrix.","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-fd8157a9d3355b90","kind":"fingerprint_related","node_a_id":"n-b1a66fdc3ca86866","node_b_id":"n-22dc599684e9805b","fingerprint":"f6-402c4469f5466095","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-402c4469f5466095"},"other_id":"n-b1a66fdc3ca86866","other":{"id":"n-b1a66fdc3ca86866","name":"mobius_x_over_x_plus_one of one_plus_derivative of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-402c4469f5466095","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","5"],["16","41","26"]],"card_rows":[["1"],["4","5"],["16","41","26"],["64","253","329","141"]],"row_sums_preview":["1","9","83","787","7697","77865","816627","8893475"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["D+1","x \u21a6 x/(x+1)"],"edges":["e-71c634e76c593d53","e-0de59c78951a358d"],"text":"A143495 \u2192 D+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A143495 \u2192 D+1 \u2192 x \u21a6 x/(x+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:      1;\n  1:      4,       5;\n  2:     16,      41,       26;\n  3:     64,     253,      329,      141;\n  4:    256,    1393,     2794,     2455,       799;\n  5:   1024,    7221,    19919,    26976,     17989,      4736;\n  6:   4096,   36105,   128836,   239624,    246011,    132584,     29371;\n  7:  16384,  176429,   784545,  1883351,   2650864,   2196461,    994944,    190497;\n  8:  65536,  849377,  4592582, 13693973,  24817591,  28139929,  19568536,   7651985,  1291020;\n  9: 262144, 4050277, 26181827, 94504754, 212067155, 308877451, 293394499, 175868872, 60527003, 9131275;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (4-(n-10)*x-(2*n-7)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[((4 - (n - 10)*x - (2*n - 7)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];\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] = ((4 - (n - 10)*x - (2*n - 7)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 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-8d887bfc40d036b9","steps":["basis A008277","x \u21a6 x/(x+1)"],"edges":["e-6d94a10ae1f2ed2c","e-a1fa34e479f265c3"],"text":"A027465 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A027465 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)","chain":null}