{"id":"n-57cc3445e053f3c8","revision":1,"first_index":1,"fingerprint":"f6-ba43151883801832","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","3"],["8","18","9"],["21","75","81","27"],["55","270","459","324","81"],["144","900","2106","2322","1215","243"],["377","2862","8559","12960","10530","4374","729"],["987","8817","32157","62235","69255","44469","15309","2187"],["2584","26550","114363","270216","383940","336798","178605","52488","6561"],["6765","78585","390582","1091502","1893213","2117745","1530900","691092","177147","19683"],["17711","229500","1293138","4176036","8570205","11696076","10767330","6613488","2598156","590490","59049"],["46368","663048","4177332","15316020","36345510","58681098","65859318","51451362","27457785","9546255","1948617","177147"],["121393","1898748","13228002","54304020","146414385","273383748","362853918","345318552","234129285","110421630","34425567","6377292","531441"],["317811","5397375","41202918","187280046","565809705","1200713301","1842853680","2073066048","1710885726","1024106490","432592974","122231430","20726199","1594323"],["832040","15246810","126572877","631144872","2112854868","5026185270","8767847025","11402440632","11117726037","8094555018","4335672825","1658095920","428341446","66961566","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","35","204","1189","6930","40391","235416","1372105","7997214","46611179","271669860","1583407981","9228778026","53789260175"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167080,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-a4eb33b32804b1a0"]},{"id":167081,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-a4eb33b32804b1a0"]},{"id":167082,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-a4eb33b32804b1a0"]},{"id":167083,"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-a4eb33b32804b1a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-a4eb33b32804b1a0","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-57cc3445e053f3c8","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":"29e1c2eaedb8248a512006d657870a38759844f2524b72450a38c05fdda514fb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-976cdb6e4a693ec8","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-e84e39d2ac132293","kind":"oeis_match","node_a_id":"n-57cc3445e053f3c8","node_b_id":null,"fingerprint":null,"oeis_id":"A001109","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001109","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001109","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    3,     3;\n   3:    8,    18,      9;\n   4:   21,    75,     81,      27;\n   5:   55,   270,    459,     324,      81;\n   6:  144,   900,   2106,    2322,    1215,     243;\n   7:  377,  2862,   8559,   12960,   10530,    4374,     729;\n   8:  987,  8817,  32157,   62235,   69255,   44469,   15309,   2187;\n   9: 2584, 26550, 114363,  270216,  383940,  336798,  178605,  52488,   6561;\n  10: 6765, 78585, 390582, 1091502, 1893213, 2117745, 1530900, 691092, 177147, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 3*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (3 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["basis A013610"],"edges":["e-a4eb33b32804b1a0"],"text":"A078812 \u2192 basis A013610"},"headline":"A078812 \u2192 basis A013610","chain":null}
