{"id":"n-0bec1a9f266d1c7a","revision":0,"first_index":0,"fingerprint":"f6-25152b6183118bbf","name":"basis_map_A013610 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.","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"],["2","3"],["6","18","9"],["16","72","90","27"],["48","288","576","432","81"],["160","1200","3312","4104","2106","243"],["576","5184","18288","31968","28188","10692","729"],["2176","22848","98208","222480","281880","191484","56862","2187"],["8448","101376","513792","1430784","2371680","2348352","1294704","314928","6561"],["33280","449280","2621952","8660736","17744832","23133600","18779040","8713008","1797714","19683"],["132096","1981440","13079808","49959936","121855104","197043840","211468320","145426752","58287924","10471356","59049"],["526336","8684544","63954432","277178112","783758592","1512120960","2017358784","1838969568","1097051688","387282708","61765254","177147"],["2101248","37822464","307316736","1489397760","4787838720","10726401024","17099050752","19404603648","15370061040","8096798880","2555168328","367048584","531441"],["8396800","163737600","1454764032","7793584128","28058088960","71552574720","132594485760","179972534016","177637657824","124382388240","58666362480","16743934440","2190599802","1594323"],["33570816","704987136","6798127104","39882792960","158910833664","454309751808","959132959488","1514496310272","1789124863680","1562094528192","980014050576","418458060576","109032561324","13105335060","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","33","205","1425","11125","95625","878125","8390625","81953125","809765625","8048828125","80244140625","801220703125","8006103515625"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","3"],["6","18","9"],["16","72","90","27"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["24","72","54"],["-8","-24","-18"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-18","-27"],["6","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201896,"position":0,"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-f4965541eafa2082"]},{"id":201897,"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-f4965541eafa2082"]},{"id":201898,"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-f4965541eafa2082"]},{"id":201899,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-f4965541eafa2082"]}],"region_exact":{},"incoming_edges":[{"id":"e-f4965541eafa2082","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-0bec1a9f266d1c7a","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":"273165c318955f1904b42d0250243dda35340718155eeab01995cff7d8d11fed","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-02236a48f5253afe","role":"sequence","name":"Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.","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:     2,      3;\n  2:     6,     18,       9;\n  3:    16,     72,      90,      27;\n  4:    48,    288,     576,     432,       81;\n  5:   160,   1200,    3312,    4104,     2106,      243;\n  6:   576,   5184,   18288,   31968,    28188,    10692,      729;\n  7:  2176,  22848,   98208,  222480,   281880,   191484,    56862,    2187;\n  8:  8448, 101376,  513792, 1430784,  2371680,  2348352,  1294704,  314928,    6561;\n  9: 33280, 449280, 2621952, 8660736, 17744832, 23133600, 18779040, 8713008, 1797714, 19683;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(24*n-72)*x+(18*n-54)*x^2)*P(n-2,x) + (6*n-18+(9*n-27)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 6 + 18*x + 9*x^2, P(3,x) = 16 + 72*x + 90*x^2 + 27*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[2, x_] := 6 + 18*x + 9*x^2;\nP[3, x_] := 16 + 72*x + 90*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (24*n - 72)*x + (18*n - 54)*x^2)*P[n - 2, x] + (6*n - 18 + (9*n - 27)*x)*P[n - 1, x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; v[3] = 6 + 18*x + 9*x^2; v[4] = 16 + 72*x + 90*x^2 + 27*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (24*n - 72)*x + (18*n - 54)*x^2)*v[i - 2] + (6*n - 18 + (9*n - 27)*x)*v[i - 1])/(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-02236a48f5253afe","steps":["basis A013610"],"edges":["e-f4965541eafa2082"],"text":"A146986 \u2192 basis A013610"},"headline":"A146986 \u2192 basis A013610","chain":null}
