{"id":"n-2fe6a26f73f75da6","revision":1,"first_index":0,"fingerprint":"f6-f61ab9580d43f2f7","name":"basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["9","12","4"],["18","42","32","8"],["36","120","148","80","16"],["72","312","536","456","192","32"],["144","768","1696","1984","1296","448","64"],["288","1824","4928","7360","6560","3488","1024","128"],["576","4224","13504","24576","27840","20096","9024","2304","256"],["1152","9600","35456","76160","104832","95872","58240","22656","5120","512"],["2304","21504","90112","223232","361984","401408","308224","161792","55552","11264","1024"],["4608","47616","223232","626688","1170432","1526784","1419264","940032","434688","133632","24576","2048"],["9216","104448","541696","1699840","3594240","5394432","5892096","4718592","2749440","1136640","316416","53248","4096"],["18432","227328","1292288","4483072","10588160","17977344","22573056","21221376","14936064","7772160","2906112","739328","114688","8192"],["36864","491520","3039232","11550720","30142464","57131008","81100800","87588864","72314880","45416448","21356544","7290880","1708032","245760","16384"],["73728","1056768","7061504","29179904","83386368","174546944","276463616","337379328","319807488","235462656","133545984","57294848","17997824","3907584","524288","32768"],["147456","2260992","16236544","72482816","225132544","515866624","902021120","1227685888","1314373632","1110540288","738017280","381681664","150585344","43810816","8863744","1114112","65536"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["25","100","400","1600","6400","25600","102400","409600","1638400","6553600","26214400","104857600","419430400","1677721600","6710886400"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["9","12","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166377,"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-8866dbf598c953fc"]},{"id":166378,"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-8866dbf598c953fc"]},{"id":166379,"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-8866dbf598c953fc"]}],"region_exact":{},"incoming_edges":[{"id":"e-8866dbf598c953fc","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2fe6a26f73f75da6","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":"b5bda7912ae22be960137142411e020e4db4df80c2e3c56b2766d0f6a9cdb4f1","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-f48bed27dbc29c5f","role":"sequence","name":"The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=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":[{"id":"l-ff1af04b7971aae0","kind":"oeis_match","node_a_id":"n-2fe6a26f73f75da6","node_b_id":null,"fingerprint":null,"oeis_id":"A309779","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A309779","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A309779","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-2fe6a26f73f75da6","oeis_id":"A309779","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":13254}],"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:    9,    12,      4;\n  1:   18,    42,     32,      8;\n  2:   36,   120,    148,     80,      16;\n  3:   72,   312,    536,    456,     192,      32;\n  4:  144,   768,   1696,   1984,    1296,     448,      64;\n  5:  288,  1824,   4928,   7360,    6560,    3488,    1024,    128;\n  6:  576,  4224,  13504,  24576,   27840,   20096,    9024,   2304,    256;\n  7: 1152,  9600,  35456,  76160,  104832,   95872,   58240,  22656,   5120,    512;\n  8: 2304, 21504,  90112, 223232,  361984,  401408,  308224, 161792,  55552,  11264,  1024;\n  9: 4608, 47616, 223232, 626688, 1170432, 1526784, 1419264, 940032, 434688, 133632, 24576, 2048;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 12*x + 4*x^2.","mathematica":"P[0, x_] := 9 + 12*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*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] = 9 + 12*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 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-f48bed27dbc29c5f","steps":["basis A013609"],"edges":["e-8866dbf598c953fc"],"text":"A391533 \u2192 basis A013609"},"headline":"A391533 \u2192 basis A013609","chain":null}