{"id":"n-fdc0c7cc2fd52cf9","revision":0,"first_index":0,"fingerprint":"f6-f2b7ac8f0752ccba","name":"shift_x_plus_one of Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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","1"],["9","7","1"],["27","34","12","1"],["81","142","83","18","1"],["243","547","451","165","25","1"],["729","2005","2142","1117","290","33","1"],["2187","7108","9341","6357","2373","469","42","1"],["6561","24604","38410","32328","15843","4536","714","52","1"],["19683","83653","151314","151930","92497","34947","8022","1038","63","1"],["59049","280483","576959","673530","490925","231317","70389","13362","1455","75","1"],["177147","930022","2144218","2854653","2426165","1372856","522786","132099","21219","1980","88","1"],["531441","3055786","7805541","11675098","11343642","7503498","3440813","1091398","234234","32406","2629","102","1"],["1594323","9964519","27933869","46384839","50738092","38449560","20657157","7909707","2136992","396396","47905","3419","117","1"],["4782969","32285041","98549092","179909391","218860285","187057906","115351173","51853011","16949673","3968250","645073","68887","4368","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","17","74","325","1432","6317","27878","123049","543148","2397545","10583234","46716589","206216896","910285253"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","-3","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174360,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-474f76b10e47b50d"]},{"id":174361,"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-11861b1a4804af97","e-474f76b10e47b50d"]},{"id":174362,"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-11861b1a4804af97","e-474f76b10e47b50d"]},{"id":174363,"position":3,"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-11861b1a4804af97"]},{"id":174364,"position":4,"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-11861b1a4804af97"]}],"region_exact":{},"incoming_edges":[{"id":"e-11861b1a4804af97","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-fdc0c7cc2fd52cf9","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":"de9b44e25d30f8f6b9217354017233711d11beb047dd7469c1104be9b1f23921","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-9fa98aec39d32ffa","role":"sequence","name":"Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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]"}]},{"id":"e-474f76b10e47b50d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fdc0c7cc2fd52cf9","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"54bf216e3eb72f13c0e52fd134a82e95364e414a6bdfca494f848c2bcd0dd33d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-9fa98aec39d32ffa","role":"sequence","name":"Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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-7ca8243202aaf21f","kind":"fingerprint_related","node_a_id":"n-fdc0c7cc2fd52cf9","node_b_id":"n-d697df4adfdac804","fingerprint":"f6-f2b7ac8f0752ccba","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f2b7ac8f0752ccba"},"other_id":"n-d697df4adfdac804","other":{"id":"n-d697df4adfdac804","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.","first_index":0,"fingerprint":"f6-f2b7ac8f0752ccba","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","7","9"]],"card_rows":[["1"],["1","3"],["1","7","9"],["1","12","34","27"]],"row_sums_preview":["1","4","17","74","325","1432","6317","27878"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1a7dcc8610d7e32","steps":["x \u21a6 x/(x+1)"],"edges":["e-88c2477d26465b0a"],"text":"A330381 \u2192 x \u21a6 x/(x+1)"},"headline":"A330381 \u2192 x \u21a6 x/(x+1)","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:     1;\n  1:     3,     1;\n  2:     9,     7,      1;\n  3:    27,    34,     12,      1;\n  4:    81,   142,     83,     18,     1;\n  5:   243,   547,    451,    165,    25,     1;\n  6:   729,  2005,   2142,   1117,   290,    33,    1;\n  7:  2187,  7108,   9341,   6357,  2373,   469,   42,    1;\n  8:  6561, 24604,  38410,  32328, 15843,  4536,  714,   52,  1;\n  9: 19683, 83653, 151314, 151930, 92497, 34947, 8022, 1038, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3+3*x+x^2)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-(3 + 3*x + x^2)*P[n - 2, x] + (4 + 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] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 3*x + x^2)*v[i - 2] + (4 + 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-9fa98aec39d32ffa","steps":["basis A007318"],"edges":["e-11861b1a4804af97"],"text":"A104029 \u2192 basis A007318"},"headline":"A104029 \u2192 basis A007318","chain":null}