{"id":"n-d07b286c67e2d8fc","revision":1,"first_index":0,"fingerprint":"f6-cb133f75685c9e6e","name":"basis_map_A013610 of Pascal-(1,4,1) array.","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"],["2","3"],["8","24","9"],["24","108","126","27"],["80","480","900","540","81"],["256","1920","5040","5400","2106","243"],["832","7488","25200","38880","27216","7776","729"],["2688","28224","116064","235440","244296","123444","27702","2187"],["8704","104448","507456","1276992","1770336","1337472","521964","96228","6561"],["28160","380160","2131200","6410880","11174112","11411280","6648480","2099520","328050","19683"],["91136","1367040","8686080","30412800","63986112","82938816","65668320","30792960","8135640","1102248","59049"],["294912","4866048","34560000","138101760","340837632","537570432","543682368","346770720","135156600","30626748","3661038","177147"],["954368","17178624","134866944","605767680","1716878592","3197242368","3958015104","3235780224","1712683440","568759968","112665492","12045996","531441"],["3088384","60223488","517902336","2583263232","8271631872","17785453824","26133331968","26256877056","17834582592","8019956448","2313776016","406729512","39326634","1594323"],["9994240","209879040","1962086400","10761707520","38429779968","93777315840","159821994240","191353052160","160119892800","92370901824","35963202960","9154956960","1445519520","127545840","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","41","285","2081","14965","108121","780045","5630161","40631525","293240201","2116305405","15273370241","110227737685","795512612281"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167157,"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-3809dca1a6c35182"]},{"id":167158,"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-3809dca1a6c35182"]},{"id":167159,"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-3809dca1a6c35182"]},{"id":167160,"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-3809dca1a6c35182"]}],"region_exact":{},"incoming_edges":[{"id":"e-3809dca1a6c35182","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-d07b286c67e2d8fc","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":"baecd92d047b81d059d032da3ef448f5cd99715b5486d03b92c0bb600579afe7","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) array.","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":[{"id":"l-f255e6029690cd9a","kind":"oeis_match","node_a_id":"n-d07b286c67e2d8fc","node_b_id":null,"fingerprint":null,"oeis_id":"A155455","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A155455","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A155455","relation":"match_row_sums"}],"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:     2,      3;\n  2:     8,     24,       9;\n  3:    24,    108,     126,      27;\n  4:    80,    480,     900,     540,       81;\n  5:   256,   1920,    5040,    5400,     2106,      243;\n  6:   832,   7488,   25200,   38880,    27216,     7776,     729;\n  7:  2688,  28224,  116064,  235440,   244296,   123444,   27702,    2187;\n  8:  8704, 104448,  507456, 1276992,  1770336,  1337472,  521964,   96228,   6561;\n  9: 28160, 380160, 2131200, 6410880, 11174112, 11411280, 6648480, 2099520, 328050, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+12*x)*P(n-2,x) + (2+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 12*x)*P[n - 2, x] + (2 + 3*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] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 12*x)*v[i - 2] + (2 + 3*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-ed9434bfd889c409","steps":["basis A013610"],"edges":["e-3809dca1a6c35182"],"text":"A081579 \u2192 basis A013610"},"headline":"A081579 \u2192 basis A013610","chain":null}
