{"id":"n-754501ecb561c326","revision":1,"first_index":0,"fingerprint":"f6-67d96af909b5ec1e","name":"shift_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"],["5","3"],["16","18","5"],["44","72","39","7"],["112","240","192","68","9"],["272","720","760","400","105","11"],["640","2016","2640","1840","720","150","13"],["1472","5376","8400","7280","3780","1176","203","15"],["3328","13824","25088","25984","16800","6944","1792","264","17"],["7424","34560","71424","86016","66528","34272","11760","2592","333","19"],["16384","84480","195840","268800","241920","149184","63840","18720","3600","410","21"],["35840","202752","520960","802560","823680","591360","303072","110880","28380","4840","495","23"],["77824","479232","1351680","2309120","2661120","2179584","1300992","570240","182160","41360","6336","588","25"],["167936","1118208","3434496","6443008","8236800","7577856","5161728","2635776","1009008","286000","58344","8112","689","27"],["360448","2580480","8572928","17518592","24600576","25113088","19219200","11202048","4996992","1697696","432432","80080","10192","798","29"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","39","162","621","2268","8019","27702","94041","314928","1043199","3424842","11160261","36137988","116385579"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","3"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","7","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163452,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-a55e8a9fee490043"]},{"id":163453,"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-a55e8a9fee490043"]},{"id":163454,"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-a55e8a9fee490043"]}],"region_exact":{},"incoming_edges":[{"id":"e-a55e8a9fee490043","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-754501ecb561c326","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"0d39b886158d806b9dc2a34cd7feff7bc7721e6cf531e62850e6f82db2236e4f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"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:    5,     3;\n  2:   16,    18,     5;\n  3:   44,    72,    39,     7;\n  4:  112,   240,   192,    68,     9;\n  5:  272,   720,   760,   400,   105,    11;\n  6:  640,  2016,  2640,  1840,   720,   150,    13;\n  7: 1472,  5376,  8400,  7280,  3780,  1176,   203,   15;\n  8: 3328, 13824, 25088, 25984, 16800,  6944,  1792,  264,  17;\n  9: 7424, 34560, 71424, 86016, 66528, 34272, 11760, 2592, 333, 19;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (n+4+3*x)*P(n-1,x) + (6+7*x+2*x^2)*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[((n + 4 + 3*x)*P[n - 1, x] + (6 + 7*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\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] = ((n + 4 + 3*x)*v[i - 1] + (6 + 7*x + 2*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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+1"],"edges":["e-dd616c2789f09167","e-a55e8a9fee490043"],"text":"A007318 \u2192 \u03b8+1+n \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u03b8+1+n \u2192 x \u21a6 x+1","chain":null}