{"id":"n-425b7d249d5598a0","revision":0,"first_index":0,"fingerprint":"f6-170b0ec849496f58","name":"basis_map_A013610 of veronese_section_2_1 of Triangle of coefficients in expansion of (1+2*x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["2"],["4"],["14","24"],["40","96"],["122","432","288"],["364","1632","1728"],["1094","6024","9504","3456"],["3280","21312","43776","27648"],["9842","73824","188352","179712","41472"],["29524","250944","763776","967680","414720"],["88574","841464","2978496","4734720","3110400","497664"],["265720","2790048","11259648","21537792","19077120","5971968"],["797162","9167376","41552352","93021696","104302080","50761728","5971968"],["2391484","29893536","150364224","385772544","524289024","348364800","83607552"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","2","4","38","136","842","3724","20078","96016","493202","2426644","12251318","60902296","305574362","1524683164"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201804,"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-de87952605a50b5f"]},{"id":201805,"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-de87952605a50b5f"]},{"id":201806,"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-de87952605a50b5f"]},{"id":201807,"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-de87952605a50b5f"]}],"region_exact":{},"incoming_edges":[{"id":"e-de87952605a50b5f","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-425b7d249d5598a0","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":"f808a6137f5db00b17936c0acde916a4276bacbec5bd015387563afba5656187","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-1864d83319c9138c","role":"sequence","name":"veronese_section_2_1 of Triangle of coefficients in expansion of (1+2*x)^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-0665f0030eb940a0","kind":"fingerprint_related","node_a_id":"n-425b7d249d5598a0","node_b_id":"n-e5bba1ecb27f003f","fingerprint":"f6-170b0ec849496f58","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-170b0ec849496f58"},"other_id":"n-e5bba1ecb27f003f","other":{"id":"n-e5bba1ecb27f003f","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","first_index":0,"fingerprint":"f6-170b0ec849496f58","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["7","12"]],"card_rows":[["1"],["2"],["7","12"],["20","48"]],"row_sums_preview":["1","2","19","68","421","1862","10039","48008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf38558d99f4fe37","steps":["basis A013610"],"edges":["e-4e6a4394ec508386"],"text":"A229032 \u2192 basis A013610"},"headline":"A229032 \u2192 basis A013610","chain":null}},{"id":"l-6fb883f0439c5958","kind":"fingerprint_related","node_a_id":"n-425b7d249d5598a0","node_b_id":"n-ac70eaa059871d64","fingerprint":"f6-170b0ec849496f58","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-170b0ec849496f58"},"other_id":"n-ac70eaa059871d64","other":{"id":"n-ac70eaa059871d64","name":"basis_map_A013610 of veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-170b0ec849496f58","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["12"]],"card_rows":[["6"],["12"],["42","72"],["120","288"]],"row_sums_preview":["0","6","12","114","408","2526","11172","60234"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-1dcfbe1f89fd593f","e-179dafd5f85bdc07"],"text":"A120909 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A120909 \u2192 Veronese 2 (j=1) \u2192 basis A013610","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:    0;\n  1:    2;\n  2:    4;\n  3:   14,    24;\n  4:   40,    96;\n  5:  122,   432,    288;\n  6:  364,  1632,   1728;\n  7: 1094,  6024,   9504,   3456;\n  8: 3280, 21312,  43776,  27648;\n  9: 9842, 73824, 188352, 179712, 41472;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+12*x)*P(n-2,x) + 2*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[(3 + 12*x)*P[n - 2, x] + 2*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (3 + 12*x)*v[i - 2] + 2*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-71965e79e47a4526","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-c6792048648d3d6d","e-de87952605a50b5f"],"text":"A013609 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A013609 \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}
