{"id":"n-1b54107fceb8b76a","revision":1,"first_index":0,"fingerprint":"f6-5d48e366590c5844","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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"],["4","1"],["16","13","1"],["64","105","27","1"],["256","729","367","46","1"],["1024","4717","3907","945","70","1"],["4096","29365","36779","14286","2025","99","1"],["16384","178641","322159","183661","41230","3836","133","1"],["65536","1071153","2695247","2139094","692901","101178","6650","172","1"],["262144","6362965","21863339","23339049","10325534","2151663","220542","10782","216","1"],["1048576","37565389","173572659","243317934","141829185","39869697","5786739","439104","16590","265","1"],["4194304","220864377","1356907655","2454593557","1838140862","671773698","130393879","13930323","813912","24475","319","1"],["16777216","1294960905","10488954615","24164460790","22820812301","10564590472","2641131004","374920172","30703827","1423675","34881","378","1"],["67108864","7578261181","80407140675","233525772465","274229352846","157753687445","49465445824","8981345152","972477792","62979202","2373657","48295","442","1"],["268435456","44292046693","612556169371","2224852124910","3213030602121","2263343796623","872842484437","197324722432","27182267632","2318295122","121684420","3801070","65247","511","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","30","197","1399","10664","86651","746045","6771932","64536235","643446139","6691637362","72398770237","813045993841","9458136496046"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","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":162399,"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-aa2cd9c4c2871bb6"]},{"id":162400,"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-aa2cd9c4c2871bb6"]},{"id":162401,"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-aa2cd9c4c2871bb6"]}],"region_exact":{},"incoming_edges":[{"id":"e-aa2cd9c4c2871bb6","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-1b54107fceb8b76a","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":"39ebf99e15b1992d34f973109aedbd3eb7b1c54c666d8865806e7831125d4686","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-c30cb06ad75c5e21","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:      4,       1;\n  2:     16,      13,        1;\n  3:     64,     105,       27,        1;\n  4:    256,     729,      367,       46,        1;\n  5:   1024,    4717,     3907,      945,       70,       1;\n  6:   4096,   29365,    36779,    14286,     2025,      99,      1;\n  7:  16384,  178641,   322159,   183661,    41230,    3836,    133,     1;\n  8:  65536, 1071153,  2695247,  2139094,   692901,  101178,   6650,   172,   1;\n  9: 262144, 6362965, 21863339, 23339049, 10325534, 2151663, 220542, 10782, 216, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(x+x^2)*P(n-2,x) - (x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (4+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) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-(x + x^2)*P[n - 2, x] - (x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (4 + 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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(x + x^2)*v[i - 2] - (x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (4 + 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-c30cb06ad75c5e21","steps":["basis A008277"],"edges":["e-aa2cd9c4c2871bb6"],"text":"A103327 \u2192 basis A008277"},"headline":"A103327 \u2192 basis A008277","chain":null}