{"id":"n-3ceccdc555ef04b6","revision":0,"first_index":0,"fingerprint":"f6-9fbd3b0e5a34e0f1","name":"basis_map_A007318 of (4,1) Pascal triangle.","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"],["5","1"],["10","7","1"],["20","24","9","1"],["40","68","42","11","1"],["80","176","152","64","13","1"],["160","432","480","280","90","15","1"],["320","1024","1392","1040","460","120","17","1"],["640","2368","3808","3472","1960","700","154","19","1"],["1280","5376","9984","10752","7392","3360","1008","192","21","1"],["2560","12032","25344","31488","25536","14112","5376","1392","234","23","1"],["5120","26624","62720","88320","82560","53760","24864","8160","1860","280","25","1"],["10240","58368","152064","239360","253440","190080","103488","41184","11880","2420","330","27","1"],["20480","126976","362496","630784","746240","633600","397056","185856","64944","16720","3080","384","29","1"],["40960","274432","851968","1624064","2123264","2013440","1427712","768768","315744","98384","22880","3848","442","31","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","18","54","162","486","1458","4374","13122","39366","118098","354294","1062882","3188646","9565938"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195268,"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-3578753419e4b8fd"]},{"id":195269,"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-3578753419e4b8fd"]},{"id":195270,"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-3578753419e4b8fd"]},{"id":195271,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-3578753419e4b8fd"]}],"region_exact":{},"incoming_edges":[{"id":"e-3578753419e4b8fd","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3ceccdc555ef04b6","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":"12ee2275f6981ece1aabe91482262baaa3fb03e31dac4aba011f377f2865f618","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-dc83b48b191d08a9","role":"sequence","name":"(4,1) Pascal triangle.","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,    1;\n  2:   10,    7,    1;\n  3:   20,   24,    9,     1;\n  4:   40,   68,   42,    11,    1;\n  5:   80,  176,  152,    64,   13,    1;\n  6:  160,  432,  480,   280,   90,   15,    1;\n  7:  320, 1024, 1392,  1040,  460,  120,   17,   1;\n  8:  640, 2368, 3808,  3472, 1960,  700,  154,  19,  1;\n  9: 1280, 5376, 9984, 10752, 7392, 3360, 1008, 192, 21, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1])/(n - 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-dc83b48b191d08a9","steps":["basis A007318"],"edges":["e-3578753419e4b8fd"],"text":"A093561 \u2192 basis A007318"},"headline":"A093561 \u2192 basis A007318","chain":null}