{"id":"n-f46004fe8e249122","revision":1,"first_index":0,"fingerprint":"f6-f3e4733f729912cf","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.","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":true,"rows":[["1"],["2","5"],["3","14","16"],["4","27","60","44"],["5","44","144","208","112"],["6","65","280","600","640","272"],["7","90","480","1360","2160","1824","640"],["8","119","756","2660","5600","7056","4928","1472"],["9","152","1120","4704","12320","20608","21504","12800","3328"],["10","189","1584","7728","24192","50400","69888","62208","32256","7424"],["11","230","2160","12000","43680","108864","188160","222720","172800","79360","16384"],["12","275","2860","17820","73920","214368","443520","654720","675840","464640","191488","35840"],["13","324","3696","25520","118800","392832","946176","1672704","2154240","1971200","1216512","454656","77824"],["14","377","4680","35464","183040","679536","1867008","3843840","5930496","6772480","5564416","3115008","1064960","167936"],["15","434","5824","48048","272272","1121120","3459456","8126976","14606592","19987968","20500480","15282176","7827456","2465792","360448"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","33","135","513","1863","6561","22599","76545","255879","846369","2775303","9034497","29229255","94065057"],"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","-2","-1"],["2","4","2"]]},"terms":[{"coeff":{"coeffs":[["2","13","26","21","6"],["0","-4","-14","-16","-6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","8","24","34","23","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163458,"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-2f364a7cea083c6e"]},{"id":163459,"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-2f364a7cea083c6e"]},{"id":163460,"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-2f364a7cea083c6e"]}],"region_exact":{},"incoming_edges":[{"id":"e-2f364a7cea083c6e","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-f46004fe8e249122","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":"833124ac352eeb095d4dede905b4f20aec5f3528020dce83b987ec5fee1d5a59","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-0486a9621eaf19a3","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.","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:  2,   5;\n  2:  3,  14,   16;\n  3:  4,  27,   60,   44;\n  4:  5,  44,  144,  208,   112;\n  5:  6,  65,  280,  600,   640,   272;\n  6:  7,  90,  480, 1360,  2160,  1824,   640;\n  7:  8, 119,  756, 2660,  5600,  7056,  4928,  1472;\n  8:  9, 152, 1120, 4704, 12320, 20608, 21504, 12800,  3328;\n  9: 10, 189, 1584, 7728, 24192, 50400, 69888, 62208, 32256, 7424;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (2-(4*n-13)*x-(14*n-26)*x^2-(16*n-21)*x^3-(6*n-6)*x^4)*P(n-1,x) + (1+8*x+24*x^2+34*x^3+23*x^4+6*x^5)*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[((2 - (4*n - 13)*x - (14*n - 26)*x^2 - (16*n - 21)*x^3 - (6*n - 6)*x^4)*P[n - 1, x] + (1 + 8*x + 24*x^2 + 34*x^3 + 23*x^4 + 6*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)]];\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] = ((2 - (4*n - 13)*x - (14*n - 26)*x^2 - (16*n - 21)*x^3 - (6*n - 6)*x^4)*v[i - 1] + (1 + 8*x + 24*x^2 + 34*x^3 + 23*x^4 + 6*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)); 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-c8c19eba9d68881f","steps":["\u03b8+1+n","x \u21a6 x/(x+1)"],"edges":["e-dd616c2789f09167","e-2f364a7cea083c6e"],"text":"A007318 \u2192 \u03b8+1+n \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u03b8+1+n \u2192 x \u21a6 x/(x+1)","chain":null}
