{"id":"n-5a85d34b4d10b026","revision":1,"first_index":0,"fingerprint":"f6-979d1a7536cb2870","name":"shift_x_plus_one of A natural number operator.","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"],["9","14","5"],["13","33","27","7"],["17","60","78","44","9"],["21","95","170","150","65","11"],["25","138","315","380","255","90","13"],["29","189","525","805","735","399","119","15"],["33","248","812","1512","1750","1288","588","152","17"],["37","315","1188","2604","3654","3402","2100","828","189","19"],["41","390","1665","4200","6930","7812","6090","3240","1125","230","21"],["45","473","2255","6435","12210","16170","15246","10230","4785","1485","275","23"],["49","564","2970","9460","20295","30888","34188","27720","16335","6820","1914","324","25"],["53","663","3822","13442","32175","55341","70356","66924","47619","25025","9438","2418","377","27"],["57","770","4823","18564","49049","94094","135135","147576","123123","78078","37037","12740","3003","434","29"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","28","80","208","512","1216","2816","6400","14336","31744","69632","151552","327680","704512"],"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":[["7","3"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160975,"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-8542387066f5b9c7"]},{"id":160976,"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-8542387066f5b9c7"]},{"id":160977,"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-8542387066f5b9c7"]}],"region_exact":{},"incoming_edges":[{"id":"e-8542387066f5b9c7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-5a85d34b4d10b026","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":"89d38c948508d6e82284f9302e798508db02ac0b1270f7efa7155b3d5d7ea21e","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-f4c1dee76f5a68b2","role":"sequence","name":"A natural number operator.","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-8700c802b65fd342","kind":"oeis_match","node_a_id":"n-5a85d34b4d10b026","node_b_id":null,"fingerprint":null,"oeis_id":"A130129","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A130129","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A130129","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-5a85d34b4d10b026","oeis_id":"A130129","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":29,"oeis_terms_available":29,"term_count":200,"raw_size":7449}],"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:  9,  14,    5;\n  3: 13,  33,   27,    7;\n  4: 17,  60,   78,   44,    9;\n  5: 21,  95,  170,  150,   65,   11;\n  6: 25, 138,  315,  380,  255,   90,   13;\n  7: 29, 189,  525,  805,  735,  399,  119,  15;\n  8: 33, 248,  812, 1512, 1750, 1288,  588, 152,  17;\n  9: 37, 315, 1188, 2604, 3654, 3402, 2100, 828, 189, 19;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = -(2*n-7-3*x)*P(n-1,x) + (4+6*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[(-(2*n - 7 - 3*x)*P[n - 1, x] + (4 + 6*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] = (-(2*n - 7 - 3*x)*v[i - 1] + (4 + 6*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-6fea5aff4f44fa54","steps":["2D+1+2n","x \u21a6 x+1"],"edges":["e-3fa49e62bb44b5e9","e-8542387066f5b9c7"],"text":"A023531 \u2192 2D+1+2n \u2192 x \u21a6 x+1"},"headline":"A023531 \u2192 2D+1+2n \u2192 x \u21a6 x+1","chain":null}