{"id":"n-73cdad053aa28c8c","revision":0,"first_index":0,"fingerprint":"f6-019e49ff32dabbfb","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Exponential Riordan array (1, x(1+x/2)).","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"],["1","2"],["2","7","6"],["4","21","36","20"],["10","68","171","188","76"],["26","220","735","1210","980","312"],["76","762","3150","6865","8310","5292","1384"],["232","2702","13356","36295","58520","55944","29344","6512"],["764","10096","57876","187880","377545","480648","378392","168352","32400"],["2620","38808","253476","957768","2306241","3668490","3853416","2576592","994896","168992"],["9496","155540","1138320","4900020","13734210","26182359","34369230","30666840","17796240","6063920","921184"],["35696","641212","5200800","25134780","80399220","178678269","281447628","314143500","243444960","124725040","38016704","5222208"],["140152","2736912","24346476","130423040","468488790","1188512028","2183036163","2924570484","2835619380","1940264480","889226448","245060928","30710464"],["568504","11996816","116170236","683351240","2723796790","7767235476","16299430719","25480757874","29670008940","25411666280","15558736336","6448182624","1621232704","186753920"],["2390480","54179608","567051576","3631863508","15900360476","50327547270","118746049422","212138758689","288311765742","296603110804","227338431320","125878155312","47594097824","10998511552","1171979904"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","81","513","3483","25839","202905","1693953","14821299","135937359","1297090017","12863135745","131979888459","1399264253487"],"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","13","13","4"],["-3","-10","-11","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","10","18","14","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175294,"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-6a33526662c4d20b"]},{"id":175295,"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-6a33526662c4d20b"]},{"id":175296,"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-6a33526662c4d20b"]}],"region_exact":{},"incoming_edges":[{"id":"e-6a33526662c4d20b","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-73cdad053aa28c8c","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":"2441b858127fcc5bc93d7db2479a0e5011a31d973ae32133eb5f26e78dc90f1a","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-11e2e791fe8625e3","role":"sequence","name":"basis_map_A007318 of Exponential Riordan array (1, x(1+x/2)).","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":[],"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:    1,     2;\n  2:    2,     7,      6;\n  3:    4,    21,     36,     20;\n  4:   10,    68,    171,    188,      76;\n  5:   26,   220,    735,   1210,     980,     312;\n  6:   76,   762,   3150,   6865,    8310,    5292,    1384;\n  7:  232,  2702,  13356,  36295,   58520,   55944,   29344,    6512;\n  8:  764, 10096,  57876, 187880,  377545,  480648,  378392,  168352,  32400;\n  9: 2620, 38808, 253476, 957768, 2306241, 3668490, 3853416, 2576592, 994896, 168992;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(3*n-4+(10*n-13)*x+(11*n-13)*x^2+(4*n-4)*x^3)*P(n-1,x) + (2+10*x+18*x^2+14*x^3+4*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[(-(3*n - 4 + (10*n - 13)*x + (11*n - 13)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*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] = (-(3*n - 4 + (10*n - 13)*x + (11*n - 13)*x^2 + (4*n - 4)*x^3)*v[i - 1] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*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-f6268da8d9636534","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-406172855e294551","e-6a33526662c4d20b"],"text":"A122848 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A122848 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}