{"id":"n-93e7087bb011d2fd","revision":0,"first_index":0,"fingerprint":"f6-d678eec589e6e72a","name":"basis_map_A013610 of (3,1) 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"],["4","3"],["8","18","9"],["16","60","72","27"],["32","168","324","270","81"],["64","432","1152","1512","972","243"],["128","1056","3600","6480","6480","3402","729"],["256","2496","10368","23760","32400","26244","11664","2187"],["512","5760","28224","78624","136080","149688","102060","39366","6561"],["1024","13056","73728","241920","508032","707616","653184","384912","131220","19683"],["2048","29184","186624","705024","1741824","2939328","3429216","2729376","1417176","433026","59049"],["4096","64512","460800","1969920","5598720","11104128","15676416","15746400","11022480","5117580","1417176","177147"],["8192","141312","1115136","5322240","17107200","39004416","64665216","78522048","69284160","43302600","18187092","4605822","531441"],["16384","307200","2654208","13989888","50181120","129330432","246343680","351039744","374134464","294457680","166281984","63772920","14880348","1594323"],["32768","663552","6230016","35942400","142331904","409204224","880678656","1441110528","1801388160","1711318752","1215937008","626391792","221079456","47829690","4782969"]],"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_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","3"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-3"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201510,"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-14167716692ec99a"]},{"id":201511,"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-14167716692ec99a"]},{"id":201512,"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-14167716692ec99a"]},{"id":201513,"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-14167716692ec99a"]}],"region_exact":{},"incoming_edges":[{"id":"e-14167716692ec99a","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-93e7087bb011d2fd","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":"519056656f7c80d5c011366f12fcf96529b90d2142c6702548fbed905fac5618","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-92128fbba2409faa","role":"sequence","name":"(3,1) 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:    4,     3;\n  2:    8,    18,     9;\n  3:   16,    60,    72,     27;\n  4:   32,   168,   324,    270,     81;\n  5:   64,   432,  1152,   1512,    972,    243;\n  6:  128,  1056,  3600,   6480,   6480,   3402,    729;\n  7:  256,  2496, 10368,  23760,  32400,  26244,  11664,   2187;\n  8:  512,  5760, 28224,  78624, 136080, 149688, 102060,  39366,   6561;\n  9: 1024, 13056, 73728, 241920, 508032, 707616, 653184, 384912, 131220, 19683;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (3*n - 3)*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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (3*n - 3)*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-92128fbba2409faa","steps":["basis A013610"],"edges":["e-14167716692ec99a"],"text":"A093560 \u2192 basis A013610"},"headline":"A093560 \u2192 basis A013610","chain":null}