{"id":"n-dfe248045567f938","revision":1,"first_index":1,"fingerprint":"f6-79e2479186e12230","name":"basis_map_A013609 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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","4"],["13","36","24"],["63","264","360","160"],["321","1800","3720","3360","1120"],["1683","11820","32760","44800","30240","8064"],["8989","75852","263760","483840","493920","266112","59136"],["48639","479248","2004912","4616640","6320160","5144832","2306304","439296"],["265729","2994192","14641200","40582080","69743520","76108032","51507456","19768320","3294720"],["1462563","18548820","103807440","336483840","696215520","953656704","864864000","500797440","168030720","24893440"],["8097453","114148980","719496360","2670370560","6462936480","10658583936","12131159040","9409720320","4760870400","1418926080","189190144"],["45046719","698734872","4898064600","20482381920","56773997280","109530989568","150085039104","146074705920","98970094080","44459683840","11918979072","1444724736"],["251595969","4258447128","32861300472","152877684960","477557720640","1055300133888","1691606332416","1981955948544","1684619988480","1013113221120","409218281472","99686006784","11076222976"],["1409933619","25858331772","217820421000","1115859064320","3878716121280","9660685402368","17738420603904","24311879454720","24873419450880","18762459555840","10143051190272","3721610919936","830716723200","85201715200"],["7923848253","156531277980","1429211216160","7994412840960","30605752457280","84835568353536","175583943575040","275660445265920","329894992627200","299494146631680","203043781484544","99686006784000","33505574502400","6901338931200","657270374400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","73","847","10321","129367","1651609","21360031","278905249","3668760487","48543499753","645382441711","8614382884849","115367108888311","1549456900170553"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["-2"],["1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["9","12"],["-6","-8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166655,"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-b82c219011a0bccd"]},{"id":166656,"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-b82c219011a0bccd"]},{"id":166657,"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-b82c219011a0bccd"]},{"id":166658,"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-b82c219011a0bccd"]}],"region_exact":{},"incoming_edges":[{"id":"e-b82c219011a0bccd","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-dfe248045567f938","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":"99e6337970fdfb45ced8f1cbaccdda6c6772d1f7b8b7aec601c0238543cd9218","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-4689fb1386d440ad","role":"sequence","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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-b969226bab4c33cd","kind":"oeis_match","node_a_id":"n-dfe248045567f938","node_b_id":null,"fingerprint":null,"oeis_id":"A084768","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A084768","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A084768","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-dfe248045567f938","oeis_id":"A084768","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":20,"oeis_terms_available":20,"term_count":200,"raw_size":23515}],"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,        4;\n   3:      13,       36,        24;\n   4:      63,      264,       360,       160;\n   5:     321,     1800,      3720,      3360,      1120;\n   6:    1683,    11820,     32760,     44800,     30240,      8064;\n   7:    8989,    75852,    263760,    483840,    493920,    266112,     59136;\n   8:   48639,   479248,   2004912,   4616640,   6320160,   5144832,   2306304,    439296;\n   9:  265729,  2994192,  14641200,  40582080,  69743520,  76108032,  51507456,  19768320,   3294720;\n  10: 1462563, 18548820, 103807440, 336483840, 696215520, 953656704, 864864000, 500797440, 168030720, 24893440;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - (6*n-9+(8*n-12)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2)*P[n - 2, x] - (6*n - 9 + (8*n - 12)*x)*P[n - 1, x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 4*x; for(n = 3, N, my(i = n); v[i] = ((n - 2)*v[i - 2] - (6*n - 9 + (8*n - 12)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4689fb1386d440ad","steps":["basis A013609"],"edges":["e-b82c219011a0bccd"],"text":"A063007 \u2192 basis A013609"},"headline":"A063007 \u2192 basis A013609","chain":null}
