{"id":"n-180398bbec0732d0","revision":1,"first_index":0,"fingerprint":"f6-c07053f4146a47c2","name":"shift_x_plus_one of Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","1"],["5","4"],["16","19","4"],["61","94","34"],["272","519","282","34"],["1385","3144","2256","496"],["7936","20903","18744","6272","496"],["50521","151418","162834","72992","11056"],["353792","1188947","1492286","842938","196864","11056"],["2702765","10064924","14444192","9852136","3119608","349504"],["22368256","91426347","147693060","118187320","47513160","8310312","349504"],["199360981","887296422","1593552930","1464241120","715085520","171334992","14873104"],["1903757312","9164847535","18115960722","18802839890","10823183920","3314853984","454512560","14873104"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","3","9","39","189","1107","7281","54351","448821","4085883","40533129","435847959","5045745069","62594829027"],"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":{"terms":[{"coeff":{"coeffs":[["0","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-3","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164158,"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-d6bacd912ada3ff2"]},{"id":164159,"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-d6bacd912ada3ff2"]},{"id":164160,"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-d6bacd912ada3ff2"]}],"region_exact":{},"incoming_edges":[{"id":"e-d6bacd912ada3ff2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-180398bbec0732d0","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":"61cd0a5c041dfd004a3f81106acb546080fd4f353338832b708632603ccfa79b","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-b4814297dd192d81","role":"sequence","name":"Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.","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-3fd66bab4f705135","kind":"oeis_match","node_a_id":"n-180398bbec0732d0","node_b_id":null,"fingerprint":null,"oeis_id":"A080635","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A080635","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A080635","relation":"match_row_sums"}],"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;\n  2:     2,      1;\n  3:     5,      4;\n  4:    16,     19,      4;\n  5:    61,     94,     34;\n  6:   272,    519,    282,    34;\n  7:  1385,   3144,   2256,   496;\n  8:  7936,  20903,  18744,  6272,   496;\n  9: 50521, 151418, 162834, 72992, 11056;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-1,x) - (1+3*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[(n + (n - 1)*x)*P[n - 1, x] - (1 + 3*x + 2*x^2)*D[P[n - 1, x], 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] = (n + (n - 1)*x)*v[i - 1] - (1 + 3*x + 2*x^2)*deriv(v[i - 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-ff270bcb802a85af","steps":["\u03b3 (scale 1/2) (shift +1)","x \u21a6 x+1"],"edges":["e-ea229ed6e1820773","e-d6bacd912ada3ff2"],"text":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 x \u21a6 x+1"},"headline":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 x \u21a6 x+1","chain":null}