{"id":"n-30551c14ec291558","revision":1,"first_index":0,"fingerprint":"f6-9a10453378e11448","name":"basis_map_A013609 of Riordan array (1+2x,x(1+x)).","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"],["3","2"],["4","10","4"],["7","26","28","8"],["11","58","104","72","16"],["18","120","300","344","176","32"],["29","236","760","1224","1024","416","64"],["47","450","1772","3688","4336","2848","960","128"],["76","838","3904","9976","15184","13984","7552","2176","256"],["123","1534","8252","25016","46848","55872","42176","19328","4864","512"],["199","2770","16900","59304","132016","193920","189440","120960","48128","10752","1024"],["322","4948","33760","134624","347504","607520","731200","603520","333568","117248","23552","2048"],["521","8760","66096","295248","867376","1760480","2523520","2565760","1830656","891392","280576","51200","4096"],["843","15394","127272","629584","2074624","4797760","7990720","9678720","8502784","5337088","2321408","661504","110592","8192"],["1364","26882","241676","1311568","4791664","12442240","23630720","33272960","34822400","26895360","15058944","5916672","1540096","237568","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","18","69","261","990","3753","14229","53946","204525","775413","2939814","11145681","42256485","160206498"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166291,"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-ba56344dd1f68fd7"]},{"id":166292,"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-ba56344dd1f68fd7"]},{"id":166293,"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-ba56344dd1f68fd7"]}],"region_exact":{},"incoming_edges":[{"id":"e-ba56344dd1f68fd7","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-30551c14ec291558","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":"c4e4c3c7a68f2ead4a91bab8aa47044c00e73e8dc2c16b208378f52cb30faf13","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-c27b11147fce8bfa","role":"sequence","name":"Riordan array (1+2x,x(1+x)).","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:   3,    2;\n  2:   4,   10,    4;\n  3:   7,   26,   28,     8;\n  4:  11,   58,  104,    72,    16;\n  5:  18,  120,  300,   344,   176,    32;\n  6:  29,  236,  760,  1224,  1024,   416,    64;\n  7:  47,  450, 1772,  3688,  4336,  2848,   960,   128;\n  8:  76,  838, 3904,  9976, 15184, 13984,  7552,  2176,  256;\n  9: 123, 1534, 8252, 25016, 46848, 55872, 42176, 19328, 4864, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (1+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (1 + 2*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; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + (1 + 2*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-c27b11147fce8bfa","steps":["basis A013609"],"edges":["e-ba56344dd1f68fd7"],"text":"A113214 \u2192 basis A013609"},"headline":"A113214 \u2192 basis A013609","chain":null}