{"id":"n-0aadd2fcd4b411db","revision":1,"first_index":0,"fingerprint":"f6-a2d96c6565c1911b","name":"basis_map_A008277 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["7","2"],["19","10","1"],["40","25","3"],["97","84","18","1"],["217","224","58","4"],["508","657","232","28","1"],["1159","1770","746","110","5"],["2683","4906","2563","510","40","1"],["6160","13145","8063","1916","185","6"],["14209","35452","25784","7539","975","54","1"],["32689","94192","79244","27098","4186","287","7"],["75316","250209","243616","98116","18550","1694","70","1"],["173383","659666","734028","339948","74950","8176","420","8"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","5","9","30","68","200","503","1426","3790","10703","29475","84014","237703","687572","1990579"],"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":[["3","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162204,"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-35e10c7226ff172b"]},{"id":162205,"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-35e10c7226ff172b"]},{"id":162206,"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-35e10c7226ff172b"]}],"region_exact":{},"incoming_edges":[{"id":"e-35e10c7226ff172b","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-0aadd2fcd4b411db","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":"e4adff777f842e21e556177c81ce620d63099daeb3a4dc46c1ef430381a7d7fc","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-ac04e52510a48bb5","role":"sequence","name":"Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:    7,    2;\n  3:   19,   10,    1;\n  4:   40,   25,    3;\n  5:   97,   84,   18,   1;\n  6:  217,  224,   58,   4;\n  7:  508,  657,  232,  28,  1;\n  8: 1159, 1770,  746, 110,  5;\n  9: 2683, 4906, 2563, 510, 40, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-2,x) + x*d/dx P(n-2,x) + 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[(3 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + P[n - 1, 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] = (3 + x)*v[i - 2] + x*deriv(v[i - 2], 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["x \u21a6 x+2 (shift +1)","basis A008277"],"edges":["e-7fc1b55fb0aa373a","e-35e10c7226ff172b"],"text":"A011973 \u2192 x \u21a6 x+2 (shift +1) \u2192 basis A008277"},"headline":"A011973 \u2192 x \u21a6 x+2 (shift +1) \u2192 basis A008277","chain":null}