{"id":"n-4944c63f5004e49c","revision":1,"first_index":0,"fingerprint":"f6-3f4e787ba07fb5f9","name":"basis_map_A008277 of Exponential Riordan array (1, x(1+x/2)).","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"],["1","1"],["2","4","1"],["4","16","9","1"],["10","66","64","16","1"],["26","286","430","180","25","1"],["76","1308","2866","1790","410","36","1"],["232","6280","19362","16926","5670","812","49","1"],["764","31636","133708","157248","72156","14924","1456","64","1"],["2620","166372","947688","1459228","880236","246036","34356","2424","81","1"],["9496","911688","6904276","13648900","10522660","3811752","712572","71460","3810","100","1"],["35696","5186512","51718876","129345876","124866280","56957692","13665036","1821732","137280","5720","121","1"],["140152","30570312","398212816","1245601016","1482561476","833937324","249410260","42365664","4220502","247390","8272","144","1"],["568504","186185416","3149326440","12209845440","17702535708","12085534604","4411164472","927202276","117037206","9028162","422994","11596","169","1"],["2390480","1169589424","25561878616","121936727640","213281833128","174536153792","76501204780","19477145116","3025089496","294346052","18080062","692146","15834","196","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","30","157","948","6487","49332","411957","3739042","36596715","383740822","4287275329","50798862988","635805146763"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162216,"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-140f8c82cfd93ce7"]},{"id":162217,"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-140f8c82cfd93ce7"]},{"id":162218,"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-140f8c82cfd93ce7"]}],"region_exact":{},"incoming_edges":[{"id":"e-140f8c82cfd93ce7","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-4944c63f5004e49c","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":"ed620da6428aebb00696ce4693bf89ae3b47b38e468b72d7339e839756be25c9","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-f6268da8d9636534","role":"sequence","name":"Exponential Riordan array (1, x(1+x/2)).","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:    1,      1;\n  2:    2,      4,      1;\n  3:    4,     16,      9,       1;\n  4:   10,     66,     64,      16,      1;\n  5:   26,    286,    430,     180,     25,      1;\n  6:   76,   1308,   2866,    1790,    410,     36,     1;\n  7:  232,   6280,  19362,   16926,   5670,    812,    49,    1;\n  8:  764,  31636, 133708,  157248,  72156,  14924,  1456,   64,  1;\n  9: 2620, 166372, 947688, 1459228, 880236, 246036, 34356, 2424, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+(n-1)*x)*P(n-2,x) + (n-1)*x*d/dx P(n-2,x) + (1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + (n - 1)*x)*P[n - 2, x] + (n - 1)*x*D[P[n - 2, x], x] + (1 + x)*P[n - 1, x] + 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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (n - 1)*x)*v[i - 2] + (n - 1)*x*deriv(v[i - 2], x) + (1 + x)*v[i - 1] + 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-97b51734d39ede6c","steps":["reverse","basis A008277"],"edges":["e-01d50252f4aaa594","e-140f8c82cfd93ce7"],"text":"A100861 \u2192 reverse \u2192 basis A008277"},"headline":"A100861 \u2192 reverse \u2192 basis A008277","chain":null}
