{"id":"n-5c1d345e9e06c9f1","revision":1,"first_index":0,"fingerprint":"f6-2a6a2f1001205b2b","name":"basis_map_A008277 of Even row Pascal-square read by antidiagonals.","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"],["3","2"],["8","13","3"],["21","64","34","4"],["55","286","266","70","5"],["144","1221","1787","796","125","6"],["377","5084","11072","7470","1946","203","7"],["987","20858","65340","62908","24270","4144","308","8"],["2584","84797","373651","494744","265834","66228","7980","444","9"],["6765","342760","2091814","3715232","2675400","913570","158922","14232","615","10"],["17711","1380390","11538790","27004458","25387667","11424084","2693152","345444","23892","825","11"],["46368","5546037","62985419","191688532","230859351","133255274","40906460","7050604","694155","38192","1078","12"],["121393","22248228","341221236","1336944378","2033559338","1476501445","574242241","127743408","16789201","1308450","58630","1378","13"],["317811","89160674","1838399784","9201481392","17483375492","15733306204","7599042052","2122031588","357488352","36997103","2338765","86996","1729","14"],["832040","357081901","9864766179","62688225168","147507389108","162639582520","96086295256","33006907928","6930536406","914332640","76424920","3997058","125398","2135","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","24","123","682","4079","26159","178823","1296271","9919320","79816424","673071482","5930739339","54464027956","520076498672"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-3","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162228,"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-25f492a2c157c337"]},{"id":162229,"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-25f492a2c157c337"]},{"id":162230,"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-25f492a2c157c337"]}],"region_exact":{},"incoming_edges":[{"id":"e-25f492a2c157c337","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-5c1d345e9e06c9f1","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"4f8cc56e8956d0915d727af055d6cc0b5c2a7555382636be563391608237f5d1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-79ebafb266da3b41","role":"sequence","name":"Even row Pascal-square read by antidiagonals.","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:    3,      2;\n  2:    8,     13,       3;\n  3:   21,     64,      34,       4;\n  4:   55,    286,     266,      70,       5;\n  5:  144,   1221,    1787,     796,     125,      6;\n  6:  377,   5084,   11072,    7470,    1946,    203,      7;\n  7:  987,  20858,   65340,   62908,   24270,   4144,    308,     8;\n  8: 2584,  84797,  373651,  494744,  265834,  66228,   7980,   444,   9;\n  9: 6765, 342760, 2091814, 3715232, 2675400, 913570, 158922, 14232, 615, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+3*x+x^2)*P(n-2,x) - (3*x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (3+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(1 + 3*x + x^2)*P[n - 2, x] - (3*x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (3 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 3*x + x^2)*v[i - 2] - (3*x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (3 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); 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-79ebafb266da3b41","steps":["basis A008277"],"edges":["e-25f492a2c157c337"],"text":"A172431 \u2192 basis A008277"},"headline":"A172431 \u2192 basis A008277","chain":null}