{"id":"n-59c91677863bdf29","revision":0,"first_index":0,"fingerprint":"f6-e28c909b767c0bb7","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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":false,"rows":[["1"],["5","3"],["24","30","9"],["116","219","135","27"],["560","1416","1323","540","81"],["2704","8568","10737","6642","2025","243"],["13056","49728","78192","64908","29970","7290","729"],["63040","280464","530604","552987","342630","126117","25515","2187"],["304384","1549056","3425760","4297968","3344085","1651428","505197","87480","6561"],["1469696","8420352","21314016","31272912","29299725","18170325","7455483","1950804","295245","19683"],["7096320","45199872","128867328","216502848","237287880","177218442","91307979","32035176","7322076","984150","59049"],["34264064","240178944","761586048","1441647072","1809677700","1581317883","981220149","432234306","132427224","26867295","3247695","177147"],["165441536","1265596416","4417949952","9303959808","13162312080","13175042040","9565721487","5074662312","1951805646","530653680","96781311","10628820","531441"],["798822400","6622218240","25235470080","58521035520","92124779760","103941409032","86456846313","53668411380","24857621334","8482782510","2072678949","343310886","34543665","1594323"],["3857055744","34444050432","142277124096","360260241408","624623353344","784327079808","735439624416","523039998168","283466928996","116482025700","35718149610","7924139604","1202119542","111602610","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","63","497","3920","30919","243873","1923544","15171919","119668241","943881120","7444845527","58721086529","463161524392","3653178276447"],"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":[["3"],["3"]]},"terms":[{"coeff":{"coeffs":[["36","18"],["-6","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["32","36","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-0c799dd389d7cc9b","recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177796,"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-d9f9312b5404ecd6"]},{"id":177797,"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-d9f9312b5404ecd6","e-e536cc74cc5a82fa"]},{"id":177798,"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-d9f9312b5404ecd6","e-e536cc74cc5a82fa"]},{"id":177799,"position":3,"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-e536cc74cc5a82fa"]},{"id":177800,"position":4,"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-e536cc74cc5a82fa"]}],"region_exact":{},"incoming_edges":[{"id":"e-e536cc74cc5a82fa","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-59c91677863bdf29","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":"2674ebf94bd357d02f694419ffff701f8feec8247a6799e8801922a438ba8177","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-226ddb8fc3536584","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(k,n-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]"}]},{"id":"e-0c799dd389d7cc9b","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-59c91677863bdf29","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"438355c2c6c6b16c9ba70cff9114f7211a55bcacd59c18b0521fc625aac66441","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-d9f9312b5404ecd6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-59c91677863bdf29","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":"2c575195fe684444b82a5720fa88bf1f485e586eae0f1f04a2ce93fa853d74ad","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-226ddb8fc3536584","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(k,n-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":[],"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:      24,      30,        9;\n  3:     116,     219,      135,       27;\n  4:     560,    1416,     1323,      540,       81;\n  5:    2704,    8568,    10737,     6642,     2025,      243;\n  6:   13056,   49728,    78192,    64908,    29970,     7290,     729;\n  7:   63040,  280464,   530604,   552987,   342630,   126117,   25515,    2187;\n  8:  304384, 1549056,  3425760,  4297968,  3344085,  1651428,  505197,   87480,   6561;\n  9: 1469696, 8420352, 21314016, 31272912, 29299725, 18170325, 7455483, 1950804, 295245, 19683;\n","recurrence_sentence":"The row polynomials satisfy (3*n+3)*P(n,x) = -(6*n-36-18*x)*P(n-1,x) + (32+36*x+9*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[(-(6*n - 36 - 18*x)*P[n - 1, x] + (32 + 36*x + 9*x^2)*D[P[n - 1, x], x])/(3*n + 3)]];\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] = (-(6*n - 36 - 18*x)*v[i - 1] + (32 + 36*x + 9*x^2)*deriv(v[i - 1], x))/(3*n + 3)); 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-21854f4257effb01","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-b1c4a0832f5f9447","e-d9f9312b5404ecd6"],"text":"A030528 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A030528 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}