{"id":"n-ed4b2207be4674b7","revision":0,"first_index":0,"fingerprint":"f6-d00b34d97cb36be9","name":"basis_map_A013611 of Triangle by columns: (1, 2, 0, 0, 0, ...) in every column.","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"],["3","4"],["3","16","16"],["3","28","80","64"],["3","40","192","384","256"],["3","52","352","1152","1792","1024"],["3","64","560","2560","6400","8192","4096"],["3","76","816","4800","16640","33792","36864","16384"],["3","88","1120","8064","35840","100352","172032","163840","65536"],["3","100","1472","12544","68096","243712","573440","851968","720896","262144"],["3","112","1872","18432","118272","516096","1548288","3145728","4128768","3145728","1048576"],["3","124","2320","25920","192000","989184","3612672","9338880","16711680","19660800","13631488","4194304"],["3","136","2816","35200","295680","1757184","7569408","23789568","54067200","86507520","92274688","58720256","16777216"],["3","148","3360","46464","436480","2939904","14598144","54067200","149225472","302776320","438304768","427819008","251658240","67108864"],["3","160","3952","59904","622336","4685824","26357760","112459776","365494272","899678208","1649410048","2181038080","1962934272","1073741824","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","35","175","875","4375","21875","109375","546875","2734375","13671875","68359375","341796875","1708984375","8544921875"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-4"],["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204105,"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-e131d3b9a76a780f"]},{"id":204106,"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-e131d3b9a76a780f"]},{"id":204107,"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-e131d3b9a76a780f"]},{"id":204108,"position":3,"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-e131d3b9a76a780f"]}],"region_exact":{},"incoming_edges":[{"id":"e-e131d3b9a76a780f","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-ed4b2207be4674b7","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":"da461a1563ecafa0ec4c4065473ea65bdd87a53185af44ae3f68e3db9335f7c5","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-e6872af866fa56cc","role":"sequence","name":"Triangle by columns: (1, 2, 0, 0, 0, ...) in every column.","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,   4;\n  2: 3,  16,   16;\n  3: 3,  28,   80,    64;\n  4: 3,  40,  192,   384,   256;\n  5: 3,  52,  352,  1152,  1792,   1024;\n  6: 3,  64,  560,  2560,  6400,   8192,   4096;\n  7: 3,  76,  816,  4800, 16640,  33792,  36864,  16384;\n  8: 3,  88, 1120,  8064, 35840, 100352, 172032, 163840,  65536;\n  9: 3, 100, 1472, 12544, 68096, 243712, 573440, 851968, 720896, 262144;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n-1+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (4*n - 4)*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (4*n - 4)*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-e6872af866fa56cc","steps":["basis A013611"],"edges":["e-e131d3b9a76a780f"],"text":"A156319 \u2192 basis A013611"},"headline":"A156319 \u2192 basis A013611","chain":null}