{"id":"n-6321b10e19f62245","revision":1,"first_index":1,"fingerprint":"f6-f2c232606f089033","name":"basis_map_A013609 of Riordan array (1, 3+x).","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"],["3","6"],["10","38","36"],["33","186","348","216"],["109","814","2272","2808","1296"],["360","3348","12420","22968","21168","7776"],["1189","13236","61248","150776","208224","152928","46656"],["3927","50910","282276","867624","1596432","1758240","1073088","279936"],["12970","191906","1240008","4569800","10504816","15422688","14121216","7371648","1679616"],["42837","712302","5255556","22581624","62264928","114248064","139489344","109268352","49828608","10077696"],["141481","2611774","21664300","106328024","341928944","752766080","1148923008","1200355200","821518848","332563968","60466176"],["467280","9482184","87343704","482062608","1771183152","4548649824","8331350976","10882850688","9935053056","6036539904","2196937728","362797056"],["1543321","34144968","345812064","2119906672","8760510096","25709672352","54940407040","86134859136","98324492544","79695539712","43535646720","14390949888","2176782336"],["5097243","122111574","1348614072","9091342416","41736278688","137783093760","336507905856","615592572288","843483386880","854940220416","623050195968","309143402496","93601640448","13060694016"],["16835050","434149790","5192613660","38177242480","192757214000","706997645952","1942582031872","4061840825216","6494599805184","7902065507328","7201718636544","4767193626624","2166624018432","605145489408","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","84","783","7299","68040","634257","5912433","55114668","513769311","4789267803","44644718160","416170266849","3879466556121","36163709805636"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166476,"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-9d236d9db88b150c"]},{"id":166477,"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-9d236d9db88b150c"]},{"id":166478,"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-9d236d9db88b150c"]}],"region_exact":{},"incoming_edges":[{"id":"e-9d236d9db88b150c","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-6321b10e19f62245","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"9e64b9cf3852bf32fb47be3ceca242732f190abbc0c792412f4ad40f7adcb0ac","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-5d39164c4108e043","role":"sequence","name":"Riordan array (1, 3+x).","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-cb9ad0f0e5cb9a69","kind":"oeis_match","node_a_id":"n-6321b10e19f62245","node_b_id":null,"fingerprint":null,"oeis_id":"A181353","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A181353","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A181353","relation":"match_row_sums"}],"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:     3,      6;\n   3:    10,     38,      36;\n   4:    33,    186,     348,      216;\n   5:   109,    814,    2272,     2808,     1296;\n   6:   360,   3348,   12420,    22968,    21168,      7776;\n   7:  1189,  13236,   61248,   150776,   208224,    152928,     46656;\n   8:  3927,  50910,  282276,   867624,  1596432,   1758240,   1073088,    279936;\n   9: 12970, 191906, 1240008,  4569800, 10504816,  15422688,  14121216,   7371648,  1679616;\n  10: 42837, 712302, 5255556, 22581624, 62264928, 114248064, 139489344, 109268352, 49828608, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (3 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 6*x; for(n = 3, N, my(i = n); v[i] = (1 + 2*x)*v[i - 2] + (3 + 6*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5d39164c4108e043","steps":["basis A013609"],"edges":["e-9d236d9db88b150c"],"text":"A099097 \u2192 basis A013609"},"headline":"A099097 \u2192 basis A013609","chain":null}
