{"id":"n-ca7afa7f470adc32","revision":0,"first_index":1,"fingerprint":"f6-25702fda18829c85","name":"shift_x_plus_one of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","2","1"],["2","5","4","1"],["5","16","18","8","1"],["11","45","70","50","15","1"],["22","110","221","224","116","26","1"],["43","252","616","805","595","238","42","1"],["85","568","1624","2576","2450","1400","448","64","1"],["170","1275","4167","7748","8961","6600","3025","792","93","1"],["341","2840","10455","22310","30380","27342","16240","6130","1335","130","1"],["683","6259","25652","61853","97042","103334","75460","37202","11803","2167","176","1"],["1366","13660","61694","165936","294925","363056","315112","192200","80356","21780","3410","232","1"],["2731","29588","145977","433238","860353","1202409","1210014","880776","458289","165178","38753","5226","299","1"],["5461","63714","340886","1106378","2427516","3796702","4346433","3678012","2292381","1033942","325416","66794","7826","378","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","4","12","48","192","720","2592","9216","32832","117504","421632","1513728","5432832","19491840"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","1"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","5","4","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171461,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-66b8121977dbcf00"]},{"id":171462,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-66b8121977dbcf00"]},{"id":171463,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-66b8121977dbcf00"]}],"region_exact":{},"incoming_edges":[{"id":"e-66b8121977dbcf00","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-ca7afa7f470adc32","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":"76730eec34b8166b6fe146e426507c56619179b1dae66ddd52526a469e3320c6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-e94410689654e39c","role":"sequence","name":"Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1;\n   2:   1,    1;\n   3:   1,    2,    1;\n   4:   2,    5,    4,    1;\n   5:   5,   16,   18,    8,    1;\n   6:  11,   45,   70,   50,   15,    1;\n   7:  22,  110,  221,  224,  116,   26,    1;\n   8:  43,  252,  616,  805,  595,  238,   42,   1;\n   9:  85,  568, 1624, 2576, 2450, 1400,  448,  64,  1;\n  10: 170, 1275, 4167, 7748, 8961, 6600, 3025, 792, 93, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+5*x+4*x^2+x^3)*P(n-3,x) - (3+6*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 2*x + x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 1 + 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 5*x + 4*x^2 + x^3)*P[n - 3, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = (2 + 5*x + 4*x^2 + x^3)*v[i - 3] - (3 + 6*x + 3*x^2)*v[i - 2] + (3 + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-07e2f0a7469aac14","steps":["Veronese 3 (j=1)","x \u21a6 x+1"],"edges":["e-1e55567759b4d30f","e-66b8121977dbcf00"],"text":"Veronese parent of A098172 \u2192 Veronese 3 (j=1) \u2192 x \u21a6 x+1"},"headline":"Veronese parent of A098172 \u2192 Veronese 3 (j=1) \u2192 x \u21a6 x+1","chain":null}
