{"id":"n-2627c6fc0d9f64c9","revision":1,"first_index":0,"fingerprint":"f6-1c0a3af5e7ed81f0","name":"basis_map_A013610 of Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","3"],["1","6","9"],["1","9","27","27"],["2","21","81","135","81"],["3","39","198","486","567","243"],["4","63","405","1350","2430","2187","729"],["6","108","819","3375","8100","11178","8019","2187"],["9","183","1602","7857","23490","43497","48114","28431","6561"],["13","297","2970","17010","61236","142884","214326","196830","98415","19683"],["19","480","5382","35181","147987","416745","790965","990711","774198","334611","59049"],["28","774","9603","70443","338580","1115856","2561706","4072194","4356504","2952450","1121931","177147"],["41","1233","16794","136755","739935","2794986","7532028","14526054","19781415","18403605","10983114","3720087","531441"],["60","1950","28926","258930","1557711","6636087","20540304","46609344","77196726","91644048","75228426","40035222","12223143","1594323"],["88","3072","49275","480735","3182166","15090543","52752627","137715390","268869780","388975446","408205737","299201283","143489070","39858075","4782969"],["129","4815","83052","877176","6335982","33102432","128985615","380846367","856741941","1463411367","1874569554","1759069710","1162792908","506994714","129140163","14348907"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["4","16","64","320","1536","7168","33792","159744","753664","3555328","16777216","79167488","373555200","1762656256","8317304832"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","3"],["1","6","9"],["1","9","27","27"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","6","9"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166951,"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-714d743cdbf8af98"]},{"id":166952,"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-714d743cdbf8af98"]},{"id":166953,"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-714d743cdbf8af98"]}],"region_exact":{},"incoming_edges":[{"id":"e-714d743cdbf8af98","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-2627c6fc0d9f64c9","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":"17f192fe9e4d6eb30431c15c344b85f9dc647aa2b6a6d2bae95b2b9fd82a1b89","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-67a75673619cc276","role":"sequence","name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","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,   3;\n  1:  1,   6,    9;\n  2:  1,   9,   27,    27;\n  3:  2,  21,   81,   135,     81;\n  4:  3,  39,  198,   486,    567,    243;\n  5:  4,  63,  405,  1350,   2430,   2187,    729;\n  6:  6, 108,  819,  3375,   8100,  11178,   8019,   2187;\n  7:  9, 183, 1602,  7857,  23490,  43497,  48114,  28431,   6561;\n  8: 13, 297, 2970, 17010,  61236, 142884, 214326, 196830,  98415,  19683;\n  9: 19, 480, 5382, 35181, 147987, 416745, 790965, 990711, 774198, 334611, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+6*x+9*x^2)*P(n-3,x) + (1+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1 + 3*x, P(1,x) = 1 + 6*x + 9*x^2, P(2,x) = 1 + 9*x + 27*x^2 + 27*x^3.","mathematica":"P[0, x_] := 1 + 3*x;\nP[1, x_] := 1 + 6*x + 9*x^2;\nP[2, x_] := 1 + 9*x + 27*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + 6*x + 9*x^2)*P[n - 3, x] + (1 + 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 + 3*x; v[2] = 1 + 6*x + 9*x^2; v[3] = 1 + 9*x + 27*x^2 + 27*x^3; for(n = 3, N, my(i = n + 1); v[i] = (1 + 6*x + 9*x^2)*v[i - 3] + (1 + 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-0d55e0bb3c384646","steps":["reverse","basis A013610"],"edges":["e-0e9e334768ea0a35","e-714d743cdbf8af98"],"text":"A102547 \u2192 reverse \u2192 basis A013610"},"headline":"A102547 \u2192 reverse \u2192 basis A013610","chain":null}