{"id":"n-9eb1bb222a47d2ea","revision":0,"first_index":0,"fingerprint":"f6-df938e0c3ee60fbd","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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"],["9","6"],["117","159","54"],["1989","4107","2826","648"],["41769","116082","120951","55998","9720"],["1044225","3653892","5113251","3577068","1250964","174960"],["30282525","127891593","225016011","211113243","111396384","31344084","3674160"],["999323325","4947203169","10494926595","12367121283","8742798504","3707867988","873505296","88179840"],["36974963025","210028247028","521886769578","740946505536","657395819289","373246675164","132432131628","26847297072","2380855680"],["1515973484025","9720407018898","27698204963538","46035409814316","49181623756929","35024988260394","16627117651668","5073703128792","903033995040","71425670400"],["68218806781125","487445440823235","1567192654982844","2985634205440974","3732341592934233","3199118055696387","1904044906918062","777011523300684","208068733026936","33014277004320","2357047123200"],["3342721532275125","26340703644459015","94340475963795816","202715011645990110","290367569449652481","291121082074755351","208466450444054970","106619181290783748","38167782757144488","9108173962181376","1304009281192320","84853696435200"],["177164241210581625","1526423432915057670","6027332668215079833","14423174179825512654","23295366635025195783","26753752558498480362","22402444074450372099","13781008175729682474","6181040556469224036","1971276747524247960","424331276428266624","55353607877566080","3309294160972800"],["10098361749003152625","94447033807002715440","407667746270691972621","1075268900315087574264","1933609213749107691099","2503369294505471603520","2400596919700607384847","1726420117143507529176","931121655099392433960","371966477980589543232","106980506152429611888","20977069259008464768","2513481298033224960","138990354760857600"],["616000066689192310125","6215695340932307509965","29117849043827332524681","83936451501401480798049","166337262552874509998919","239717941583543613914175","259088030354483272634067","213338488532281291597851","134487326232520777284480","64590429636288621665352","23264302384424735770608","6093724001658848885808","1097290475835407637120","121585070051907436800","6254565964238592000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","330","9570","344520","14814360","740718000","42220926000","2702139264000","191851887744000","14964447244032000","1271978015742720000","117021977448330240000","11585175767384693760000","1228028631342777538560000"],"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":[["5","3"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191113,"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-e90084655a406f32"]},{"id":191114,"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-ce1c10e98651a416","e-e90084655a406f32"]},{"id":191115,"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-ce1c10e98651a416","e-e90084655a406f32"]},{"id":191116,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-ce1c10e98651a416","e-e90084655a406f32"]},{"id":191117,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-ce1c10e98651a416"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce1c10e98651a416","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9eb1bb222a47d2ea","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"c5c38634be61fbc7b89360145a3a05cf585bea41abe1d907541a6f32377252bb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-70594f2875298da0","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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]"}]},{"id":"e-e90084655a406f32","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9eb1bb222a47d2ea","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":"c4c2aa4e4bc317fb0f2dab8bb484900e84248846e9bb9270d3984085d47b9ae8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-70594f2875298da0","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:             9,             6;\n  2:           117,           159,             54;\n  3:          1989,          4107,           2826,            648;\n  4:         41769,        116082,         120951,          55998,           9720;\n  5:       1044225,       3653892,        5113251,        3577068,        1250964,         174960;\n  6:      30282525,     127891593,      225016011,      211113243,      111396384,       31344084,        3674160;\n  7:     999323325,    4947203169,    10494926595,    12367121283,     8742798504,     3707867988,      873505296,      88179840;\n  8:   36974963025,  210028247028,   521886769578,   740946505536,   657395819289,   373246675164,   132432131628,   26847297072,   2380855680;\n  9: 1515973484025, 9720407018898, 27698204963538, 46035409814316, 49181623756929, 35024988260394, 16627117651668, 5073703128792, 903033995040, 71425670400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+5+(3*n+3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n + 5 + (3*n + 3)*x)*P[n - 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] = (4*n + 5 + (3*n + 3)*x)*v[i - 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-2feb6dc01f424b0a","steps":["basis A013610","basis A007318"],"edges":["e-45b4054949a52789","e-ce1c10e98651a416"],"text":"A145324 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A145324 \u2192 basis A013610 \u2192 basis A007318","chain":null}
