{"id":"n-f99f79888b04bfd9","revision":0,"first_index":0,"fingerprint":"f6-0e2849de7a670bd3","name":"reversal of basis_map_A007318 of Pascal-(1,3,1) array.","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"],["1","2"],["1","7","7"],["1","12","30","20"],["1","17","78","122","61"],["1","22","151","404","455","182"],["1","27","249","991","1863","1641","547"],["1","32","372","2008","5510","7944","5740","1640"],["1","37","520","3580","13246","27526","32140","19684","4921"],["1","42","693","5832","27546","76572","127554","125016","66429","14762"],["1","47","891","8889","51510","182142","403014","559122","471933","221435","44287"],["1","52","1114","12876","88863","385296","1079652","1977528","2347887","1739636","730730","132860"],["1","57","1362","17918","143955","744219","2551200","5892300","9189351","9528575","6290106","2391486","398581"],["1","62","1635","24140","221761","1337346","5462115","15389544","30145491","40883522","37609825","22382796","7772323","1195742"],["1","67","1933","31667","327881","2266487","10801329","36200031","86255079","146419457","175530647","145058489","78582691","25110589","3587227"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","63","279","1215","5319","23247","101655","444447","1943271","8496495","37149111","162426303","710173575"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","24","32"],["1","-3","-40","-48"],["0","5","16","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","25","88","64"],["0","-1","-18","-48","-32"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","17","32","16"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176463,"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-7c5d7e65b5d5d237"]},{"id":176464,"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-7c5d7e65b5d5d237"]},{"id":176465,"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-7c5d7e65b5d5d237"]}],"region_exact":{},"incoming_edges":[{"id":"e-7c5d7e65b5d5d237","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-f99f79888b04bfd9","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"1c3f1b0f4683ba67f57a1603ab8ba6c7d9c89b46287a2fc364c0f10805d84af7","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-bfe76bb735cd71bb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,3,1) array.","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-b49ed0b6bef47b8e","kind":"candidate_equal","node_a_id":"n-67cce3037dca960d","node_b_id":"n-f99f79888b04bfd9","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-67cce3037dca960d","other":{"id":"n-67cce3037dca960d","name":"mobius_x_over_x_plus_one of Pascal-(1,3,1) array.","first_index":0,"fingerprint":"f6-0e2849de7a670bd3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","7","7"]],"card_rows":[["1"],["1","2"],["1","7","7"],["1","12","30","20"]],"row_sums_preview":["1","3","15","63","279","1215","5319","23247"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d8f9ad9ce0e039d0","steps":["x \u21a6 x/(x+1)"],"edges":["e-923647fd37394295"],"text":"A081578 \u2192 x \u21a6 x/(x+1)"},"headline":"A081578 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-0b19439963473850","kind":"fingerprint_related","node_a_id":"n-bfe76bb735cd71bb","node_b_id":"n-f99f79888b04bfd9","fingerprint":"f6-0e2849de7a670bd3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0e2849de7a670bd3"},"other_id":"n-bfe76bb735cd71bb","other":{"id":"n-bfe76bb735cd71bb","name":"basis_map_A007318 of Pascal-(1,3,1) array.","first_index":0,"fingerprint":"f6-0e2849de7a670bd3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["7","7","1"]],"card_rows":[["1"],["2","1"],["7","7","1"],["20","30","12","1"]],"row_sums_preview":["1","3","15","63","279","1215","5319","23247"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d8f9ad9ce0e039d0","steps":["x \u21a6 x+1"],"edges":["e-118ee1d5972a9fb3"],"text":"A081578 \u2192 x \u21a6 x+1"},"headline":"A081578 \u2192 x \u21a6 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: 1,  2;\n  2: 1,  7,   7;\n  3: 1, 12,  30,   20;\n  4: 1, 17,  78,  122,    61;\n  5: 1, 22, 151,  404,   455,   182;\n  6: 1, 27, 249,  991,  1863,  1641,    547;\n  7: 1, 32, 372, 2008,  5510,  7944,   5740,   1640;\n  8: 1, 37, 520, 3580, 13246, 27526,  32140,  19684,  4921;\n  9: 1, 42, 693, 5832, 27546, 76572, 127554, 125016, 66429, 14762;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (n+(5*n^2-3*n)*x+(16*n^2-40*n+24)*x^2+(16*n^2-48*n+32)*x^3)*P(n-1,x) - (n*x+(18*n-25)*x^2+(48*n-88)*x^3+(32*n-64)*x^4)*d/dx P(n-1,x) + (x^2+17*x^3+32*x^4+16*x^5)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (5*n^2 - 3*n)*x + (16*n^2 - 40*n + 24)*x^2 + (16*n^2 - 48*n + 32)*x^3)*P[n - 1, x] - (n*x + (18*n - 25)*x^2 + (48*n - 88)*x^3 + (32*n - 64)*x^4)*D[P[n - 1, x], x] + (x^2 + 17*x^3 + 32*x^4 + 16*x^5)*D[P[n - 1, x], {x, 2}])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + (5*n^2 - 3*n)*x + (16*n^2 - 40*n + 24)*x^2 + (16*n^2 - 48*n + 32)*x^3)*v[i - 1] - (n*x + (18*n - 25)*x^2 + (48*n - 88)*x^3 + (32*n - 64)*x^4)*deriv(v[i - 1], x) + (x^2 + 17*x^3 + 32*x^4 + 16*x^5)*deriv(deriv(v[i - 1], x), x))/(n)); 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-d8f9ad9ce0e039d0","steps":["x \u21a6 x+1","reverse"],"edges":["e-118ee1d5972a9fb3","e-7c5d7e65b5d5d237"],"text":"A081578 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A081578 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}