{"id":"n-e3e112b1337439e7","revision":0,"first_index":0,"fingerprint":"f6-e75183813a6a02ba","name":"basis_map_A013611 of Triangle by columns, (3, 1, 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":[["3"],["4","12"],["4","28","48"],["4","44","160","192"],["4","60","336","832","768"],["4","76","576","2176","4096","3072"],["4","92","880","4480","12800","19456","12288"],["4","108","1248","8000","30720","70656","90112","49152"],["4","124","1680","12992","62720","193536","372736","409600","196608"],["4","140","2176","19712","114688","444416","1146880","1900544","1835008","786432"],["4","156","2736","28416","193536","903168","2924544","6488064","9437184","8126464","3145728"],["4","172","3360","39360","307200","1677312","6537216","18186240","35389440","45875200","35651584","12582912"],["4","188","4048","52800","464640","2906112","13246464","44335104","108134400","187432960","219152384","155189248","50331648"],["4","204","4800","68992","675840","4764672","24870912","97320960","285474816","619970560","968884224","1031798784","671088640","201326592"],["4","220","5616","88192","951808","7468032","43929600","196804608","674758656","1761869824","3448766464","4907335680","4798283776","2885681152","805306368"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["3","16","80","400","2000","10000","50000","250000","1250000","6250000","31250000","156250000","781250000","3906250000","19531250000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"],["4","12"]],"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":204097,"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-c38f1fdbefd2282d"]},{"id":204098,"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-c38f1fdbefd2282d"]},{"id":204099,"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-c38f1fdbefd2282d"]},{"id":204100,"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-c38f1fdbefd2282d"]}],"region_exact":{},"incoming_edges":[{"id":"e-c38f1fdbefd2282d","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-e3e112b1337439e7","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":"17683d89e5fb7e7e9849e50ed64dfb7d94b567aaec25b626e0f131ac90612c2f","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-863aa63e001c0689","role":"sequence","name":"Triangle by columns, (3, 1, 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: 3;\n  1: 4,  12;\n  2: 4,  28,   48;\n  3: 4,  44,  160,   192;\n  4: 4,  60,  336,   832,    768;\n  5: 4,  76,  576,  2176,   4096,   3072;\n  6: 4,  92,  880,  4480,  12800,  19456,   12288;\n  7: 4, 108, 1248,  8000,  30720,  70656,   90112,   49152;\n  8: 4, 124, 1680, 12992,  62720, 193536,  372736,  409600,  196608;\n  9: 4, 140, 2176, 19712, 114688, 444416, 1146880, 1900544, 1835008, 786432;\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) = 3, P(1,x) = 4 + 12*x.","mathematica":"P[0, x_] := 3;\nP[1, x_] := 4 + 12*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] = 3; v[2] = 4 + 12*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-863aa63e001c0689","steps":["basis A013611"],"edges":["e-c38f1fdbefd2282d"],"text":"A136157 \u2192 basis A013611"},"headline":"A136157 \u2192 basis A013611","chain":null}