{"id":"n-5bf7e7b307eaaec8","revision":0,"first_index":0,"fingerprint":"f6-bbe4c1ef283db3b0","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).","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","6"],["9","45","54"],["44","312","720","540"],["225","2070","7020","10395","5670"],["1182","13374","59724","131544","142884","61236"],["6321","84903","469980","1372140","2228310","1908522","673596"],["34232","532320","3514752","12772080","27585360","35411904","25019280","7505784"],["187137","3306636","25356240","110207790","296943570","507891384","538540002","323686935","84440070"],["1030490","20392710","178085160","900705960","2907564660","6212426220","8785937160","7931111760","4146945660","956987460"],["5707449","125046801","1225023426","7066232316","26577216666","68105607570","120422798496","145079298936","113977206486","52730009046","10909657044"],["31760676","763199928","8288120880","53691690780","230538639240","688898141568","1461905643744","2203159130640","2310863719320","1606694946480","666480866688","124965162504"],["177435297","4639968450","55319836572","397616764545","1918890513540","6550444476576","16218810808200","29348189632476","38520180746190","35765492342580","22299338997936","8383079651310","1437099368796"],["994551222","28116779166","365093556660","2883031295760","15447080852280","59300580250296","167812033834368","354437040822480","558746477552340","649330199193060","540717849684936","305549435027088","105018800027400","16581915793800"],["5590402785","169900981515","2386613843640","20538018507240","120961903516740","515794442892300","1642133700104040","3965405776829640","7298533288293390","10189319663527290","10621533001467960","8016961478783400","4141925680779900","1311155770266900","191876454185400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","108","1616","25380","409944","6743772","112375712","1890559764","32041187240","546223804236","9356281022448","160899280542468","2776218649220856","48038701284382140"],"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","6"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0"],["2"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","6"],["-9","-18"],["6","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202301,"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-003603dbe1f86f22"]},{"id":202302,"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-003603dbe1f86f22"]},{"id":202303,"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-003603dbe1f86f22"]},{"id":202304,"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-003603dbe1f86f22"]}],"region_exact":{},"incoming_edges":[{"id":"e-003603dbe1f86f22","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-5bf7e7b307eaaec8","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":"bea01256850bd36e8a9e6d6fe03db01bafb93e8fe7c246b0587d773078affe58","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-2fa133578d3075fe","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).","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-fa97d64b89de0316","kind":"fingerprint_related","node_a_id":"n-5bf7e7b307eaaec8","node_b_id":"n-97cba47fa866c4c1","fingerprint":"f6-bbe4c1ef283db3b0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bbe4c1ef283db3b0"},"other_id":"n-97cba47fa866c4c1","other":{"id":"n-97cba47fa866c4c1","name":"basis_map_A013610 of Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.","first_index":0,"fingerprint":"f6-bbe4c1ef283db3b0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","6"],["18","90","108"]],"card_rows":[["1"],["2","6"],["18","90","108"],["264","1872","4320","3240"]],"row_sums_preview":["1","8","216","9696","609120","49193280","4855515840","566373588480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4cba07695b3e3f90","steps":["basis A013610"],"edges":["e-43b7172d2592543b"],"text":"A357367 \u2192 basis A013610"},"headline":"A357367 \u2192 basis A013610","chain":null}}],"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:       2,        6;\n  2:       9,       45,        54;\n  3:      44,      312,       720,       540;\n  4:     225,     2070,      7020,     10395,       5670;\n  5:    1182,    13374,     59724,    131544,     142884,      61236;\n  6:    6321,    84903,    469980,   1372140,    2228310,    1908522,     673596;\n  7:   34232,   532320,   3514752,  12772080,   27585360,   35411904,   25019280,    7505784;\n  8:  187137,  3306636,  25356240, 110207790,  296943570,  507891384,  538540002,  323686935,   84440070;\n  9: 1030490, 20392710, 178085160, 900705960, 2907564660, 6212426220, 8785937160, 7931111760, 4146945660, 956987460;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n)*P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n^2-9*n+3+(12*n^2-18*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n)*P[n - 2, x] + (6*n^2 - 9*n + 3 + (12*n^2 - 18*n + 6)*x)*P[n - 1, x])/(n^2 - n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n)*v[i - 2] + (6*n^2 - 9*n + 3 + (12*n^2 - 18*n + 6)*x)*v[i - 1])/(n^2 - n)); 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-2fa133578d3075fe","steps":["basis A013610"],"edges":["e-003603dbe1f86f22"],"text":"A386789 \u2192 basis A013610"},"headline":"A386789 \u2192 basis A013610","chain":null}