{"id":"n-f124fdca65bac486","revision":0,"first_index":0,"fingerprint":"f6-59885d546c14be4d","name":"basis_map_A013609 of The (1,2)-Pascal triangle (or Lucas triangle) read by rows.","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":true,"rows":[["2"],["3","4"],["6","14","8"],["12","40","44","16"],["24","104","168","120","32"],["48","256","544","576","304","64"],["96","608","1600","2240","1760","736","128"],["192","1408","4416","7680","8000","4992","1728","256"],["384","3200","11648","24192","31360","25984","13440","3968","512"],["768","7168","29696","71680","111104","114688","78848","34816","8960","1024"],["1536","15872","73728","202752","365568","451584","387072","227328","87552","19968","2048"],["3072","34816","179200","552960","1136640","1634304","1677312","1228800","629760","215040","44032","4096"],["6144","75776","428032","1464320","3379200","5541888","6623232","5812224","3717120","1689600","518144","96256","8192"],["12288","163840","1007616","3784704","9687040","17842176","24330240","24870912","19058688","10813440","4415488","1228800","208896","16384"],["24576","352256","2342912","9584640","26943488","55058432","84344832","98402304","87859200","59744256","30457856","11288576","2875392","450560","32768"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","7","28","112","448","1792","7168","28672","114688","458752","1835008","7340032","29360128","117440512","469762048"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-2"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198641,"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-70e2c8c9921b6673"]},{"id":198642,"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-70e2c8c9921b6673"]},{"id":198643,"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-70e2c8c9921b6673"]},{"id":198644,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-70e2c8c9921b6673"]}],"region_exact":{},"incoming_edges":[{"id":"e-70e2c8c9921b6673","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-f124fdca65bac486","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":"632efd0cd07d62de8a51ce7934e88e31eb313761d9991992eaf1681bdbb53812","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b636fb6adf08770c","role":"sequence","name":"The (1,2)-Pascal triangle (or Lucas triangle) read by rows.","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-41099c31e08230ce","kind":"fingerprint_related","node_a_id":"n-14b6be09569eb319","node_b_id":"n-f124fdca65bac486","fingerprint":"f6-59885d546c14be4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-59885d546c14be4d"},"other_id":"n-14b6be09569eb319","other":{"id":"n-14b6be09569eb319","name":"basis_map_A007318 of Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.","first_index":0,"fingerprint":"f6-59885d546c14be4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["4","3"],["4","7","3"]],"card_rows":[["3"],["4","3"],["4","7","3"],["4","11","10","3"]],"row_sums_preview":["3","7","14","28","56","112","224","448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-863aa63e001c0689","steps":["basis A007318"],"edges":["e-0e6665ab2554038a"],"text":"A136157 \u2192 basis A007318"},"headline":"A136157 \u2192 basis A007318","chain":null}},{"id":"l-fb1323c2647fe194","kind":"fingerprint_related","node_a_id":"n-7548030fd3f3466d","node_b_id":"n-f124fdca65bac486","fingerprint":"f6-59885d546c14be4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-59885d546c14be4d"},"other_id":"n-7548030fd3f3466d","other":{"id":"n-7548030fd3f3466d","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-59885d546c14be4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","3"],["12","21","9"],["36","99","90","27"]],"card_rows":[["4","3"],["12","21","9"],["36","99","90","27"],["108","405","567","351","81"]],"row_sums_preview":["7","42","252","1512","9072","54432","326592","1959552"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-cf7bdd46272d1037"],"text":"A038221 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038221 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}}],"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:   2;\n  1:   3,    4;\n  2:   6,   14,     8;\n  3:  12,   40,    44,    16;\n  4:  24,  104,   168,   120,     32;\n  5:  48,  256,   544,   576,    304,     64;\n  6:  96,  608,  1600,  2240,   1760,    736,   128;\n  7: 192, 1408,  4416,  7680,   8000,   4992,  1728,   256;\n  8: 384, 3200, 11648, 24192,  31360,  25984, 13440,  3968,  512;\n  9: 768, 7168, 29696, 71680, 111104, 114688, 78848, 34816, 8960, 1024;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(2*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (2*n - 2)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (2*n - 2)*x)*v[i - 1])/(n - 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-b636fb6adf08770c","steps":["basis A013609"],"edges":["e-70e2c8c9921b6673"],"text":"A029635 \u2192 basis A013609"},"headline":"A029635 \u2192 basis A013609","chain":null}
