{"id":"n-79ed04620ce13b77","revision":0,"first_index":1,"fingerprint":"f6-b9ca97c881366b5c","name":"basis_map_A008277 of basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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"],["4","2"],["20","22","4"],["96","192","80","8"],["464","1516","1100","248","16"],["2240","11368","12856","4920","704","32"],["10816","82712","137416","79592","18976","1888","64"],["52224","590592","1390336","1145664","405184","66240","4864","128"],["252160","4165904","13568624","15308064","7540432","1802112","215104","12160","256"],["1217536","29147744","129184672","194455200","128254848","42048160","7256192","661120","29696","512"],["5878784","202821280","1208653664","2383139232","2049069888","885494368","207241280","27061632","1945600","71168","1024"],["28385280","1406006784","11166644736","28451620992","31283085696","17340244864","5287044608","928004480","94951424","5527552","167936","2048"],["137056256","9721496512","102220043328","333098199040","461655387456","321824660416","124360052672","28179761792","3848072704","316965888","15252480","391168","4096"],["661766144","67095902848","929348511104","3842271576448","6638361317888","5734185620224","2752683367040","783233987456","137076015104","14980903936","1015076864","41064448","901120","8192"],["3195289600","462502040960","8406183234176","43816834863744","93550940015104","98984177339648","58144715276672","20359534325888","4441309254656","618401311744","55323643904","3138504704","108257280","2056192","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","46","376","3344","32120","331464","3655232","42864816","532255680","6971377920","95991686400","1385377343808","20900956018816","328846365430656"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"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":196789,"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-81c4c302ec106af9"]},{"id":196790,"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-81c4c302ec106af9"]},{"id":196791,"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-81c4c302ec106af9"]}],"region_exact":{},"incoming_edges":[{"id":"e-81c4c302ec106af9","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-79ed04620ce13b77","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":"4ad0c7b5b399f43fde08d4296ca083408ebbe89e3cb79bfc36848cb5de9477a4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-726f071d833f296a","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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-91189cb07657865b","kind":"fingerprint_related","node_a_id":"n-3b34659855d92bfc","node_b_id":"n-79ed04620ce13b77","fingerprint":"f6-b9ca97c881366b5c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b9ca97c881366b5c"},"other_id":"n-3b34659855d92bfc","other":{"id":"n-3b34659855d92bfc","name":"basis_map_A008277 of basis_map_A007318 of Riordan array (1, 2+2x).","first_index":0,"fingerprint":"f6-b9ca97c881366b5c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["20","22","4"]],"card_rows":[["1"],["4","2"],["20","22","4"],["96","192","80","8"]],"row_sums_preview":["1","6","46","376","3344","32120","331464","3655232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f5a001953ee28e8f","steps":["basis A007318","basis A008277"],"edges":["e-23c3ede392db8800","e-1e8c9bb4e261b8b7"],"text":"A099040 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A099040 \u2192 basis A007318 \u2192 basis A008277","chain":null}}],"oeis":[],"bfiles":[],"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:       4,        2;\n   3:      20,       22,         4;\n   4:      96,      192,        80,         8;\n   5:     464,     1516,      1100,       248,        16;\n   6:    2240,    11368,     12856,      4920,       704,       32;\n   7:   10816,    82712,    137416,     79592,     18976,     1888,      64;\n   8:   52224,   590592,   1390336,   1145664,    405184,    66240,    4864,    128;\n   9:  252160,  4165904,  13568624,  15308064,   7540432,  1802112,  215104,  12160,   256;\n  10: 1217536, 29147744, 129184672, 194455200, 128254848, 42048160, 7256192, 661120, 29696, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = (4 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 2], 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A013609","basis A008277"],"edges":["e-081205c1b1ff787d","e-81c4c302ec106af9"],"text":"A063967 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A063967 \u2192 basis A013609 \u2192 basis A008277","chain":null}