{"id":"n-b2e2c5159c22bcc9","revision":1,"first_index":0,"fingerprint":"f6-f702baac9257fe05","name":"basis_map_A013611 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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":false,"rows":[["1"],["3","8"],["8","36","32"],["21","132","240","128"],["55","444","1216","1344","512"],["144","1420","5184","8768","6912","2048"],["377","4392","20016","45696","55296","33792","8192"],["987","13264","72432","208384","341760","320512","159744","32768"],["2584","39348","250336","869184","1803520","2294784","1753088","737280","131072"],["6765","115116","835968","3400512","8545536","13777920","14278656","9191424","3342336","524288"],["17711","333060","2718032","12676224","37435136","73221120","96194560","83951616","46661632","14942208","2097152"],["46368","954908","8650368","45500288","154464768","355625984","567189504","627441664","472449024","230948864","66060288","8388608"],["121393","2717136","27052704","158426112","607960320","1611515904","3027877888","4067131392","3880452096","2567700480","1119879168","289406976","33554432"],["317811","7682072","83376288","537988864","2303120640","6910763008","14962507776","23685464064","27437432832","22993960960","13564379136","5339348992","1258291200","134217728"],["832040","21600324","253804448","1789045632","8453357056","28333255680","69502697472","126839291904","173173702656","176163913728","131549102080","69986156544","25098715136","5435817984","536870912"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","76","521","3571","24476","167761","1149851","7881196","54018521","370248451","2537720636","17393796001","119218851371","817138163596"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167764,"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-a3f64a2b3fc7c9ce"]},{"id":167765,"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-a3f64a2b3fc7c9ce"]},{"id":167766,"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-a3f64a2b3fc7c9ce"]},{"id":167767,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-a3f64a2b3fc7c9ce"]}],"region_exact":{},"incoming_edges":[{"id":"e-a3f64a2b3fc7c9ce","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-b2e2c5159c22bcc9","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":"1abdbc177e8889f47152a5a719a358e4345f782bff84171d50a389534270fbdf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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-f0b01c23e2480dd4","kind":"oeis_match","node_a_id":"n-b2e2c5159c22bcc9","node_b_id":null,"fingerprint":null,"oeis_id":"A056914","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":"A056914","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A056914","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-b2e2c5159c22bcc9","oeis_id":"A056914","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":17667}],"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:    3,      8;\n  2:    8,     36,     32;\n  3:   21,    132,    240,     128;\n  4:   55,    444,   1216,    1344,     512;\n  5:  144,   1420,   5184,    8768,    6912,     2048;\n  6:  377,   4392,  20016,   45696,   55296,    33792,     8192;\n  7:  987,  13264,  72432,  208384,  341760,   320512,   159744,   32768;\n  8: 2584,  39348, 250336,  869184, 1803520,  2294784,  1753088,  737280,  131072;\n  9: 6765, 115116, 835968, 3400512, 8545536, 13777920, 14278656, 9191424, 3342336, 524288;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 4*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] = 1; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + 4*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-353fb5a029e4e669","steps":["basis A013611"],"edges":["e-a3f64a2b3fc7c9ce"],"text":"A207607 \u2192 basis A013611"},"headline":"A207607 \u2192 basis A013611","chain":null}