{"id":"n-f2968a7a3d3a43cd","revision":1,"first_index":0,"fingerprint":"f6-56f113efd6560c46","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (2+3x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["4","9"],["8","54"],["16","216","81"],["32","720","810"],["64","2160","4860","729"],["128","6048","22680","10206"],["256","16128","90720","81648","6561"],["512","41472","326592","489888","118098"],["1024","103680","1088640","2449440","1180980","59049"],["2048","253440","3421440","10777536","8660520","1299078"],["4096","608256","10264320","43110144","51963120","15588936","531441"],["8192","1437696","29652480","160123392","270208224","135104112","13817466"],["16384","3354624","83026944","560431872","1260971712","945728784","193444524","4782969"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","13","62","313","1562","7813","39062","195313","976562","4882813","24414062","122070313","610351562","3051757813"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_0","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","9"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168635,"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-bdf59334244a08a5"]},{"id":168636,"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-bdf59334244a08a5"]},{"id":168637,"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-bdf59334244a08a5"]}],"region_exact":{},"incoming_edges":[{"id":"e-bdf59334244a08a5","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-f2968a7a3d3a43cd","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"4095e56de615ea48b0e415fe28e17e371c74a6995a8c5853271e2ccbc170a84e","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-5733d79f45a14084","role":"sequence","name":"Triangle of coefficients in expansion of (2+3x)^n.","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-437e86c7092f0348","kind":"oeis_match","node_a_id":"n-f2968a7a3d3a43cd","node_b_id":null,"fingerprint":null,"oeis_id":"A081340","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A081340","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A081340","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-f2968a7a3d3a43cd","oeis_id":"A081340","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":14840}],"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:   2;\n  2:   4,     9;\n  3:   8,    54;\n  4:  16,   216,     81;\n  5:  32,   720,    810;\n  6:  64,  2160,   4860,    729;\n  7: 128,  6048,  22680,  10206;\n  8: 256, 16128,  90720,  81648,   6561;\n  9: 512, 41472, 326592, 489888, 118098;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (4-9*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (4 - 9*x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = 4*v[i - 1] - (4 - 9*x)*v[i - 2]); 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-5733d79f45a14084","steps":["Veronese 2"],"edges":["e-bdf59334244a08a5"],"text":"A013620 \u2192 Veronese 2"},"headline":"A013620 \u2192 Veronese 2","chain":null}