{"id":"n-a83326ab79ee2ecd","revision":0,"first_index":0,"fingerprint":"f6-2a2e8db5a042c827","name":"mobius_x_over_x_plus_one of type_b_eulerian","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","8","8"],["1","26","72","48"],["1","80","464","768","384"],["1","242","2640","8160","9600","3840"],["1","728","14168","72960","151680","138240","46080"],["1","2186","73752","595728","1948800","3037440","2257920","645120"],["1","6560","377504","4612608","22305024","52899840","65802240","41287680","10321920"],["1","19682","1913760","34553280","237646080","804948480","1490227200","1540546560","836075520","185794560"],["1","59048","9647528","253355520","2415240960","11230894080","29032335360","43971379200","38862028800","18579456000","3715891200"],["1","177146","48473832","1831373808","23764012800","147692244480","512191088640","1066023383040","1364196556800","1052526182400","449622835200","81749606400"],["1","531440","243077744","13110459648","228527105664","1862254817280","8430791086080","23161025986560","40247715594240","44553535488000","30492603187200","11771943321600","1961990553600"],["1","1594322","1217514480","93231684000","2161627628160","22770074046720","131947341926400","465446465003520","1054787580887040","1570976054968320","1531415376691200","941591966515200","331576403558400","51011754393600"],["1","4782968","6093949688","659926874880","20200502125440","272087090795520","1988556333603840","8828959762022400","25378532315136000","48834721500364800","63579244009881600","55347753517056000","30887617285324800","9998303861145600","1428329123020800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","147","1697","24483","423857","8560947","197613377","5131725123","148070287697","4699645934547","162723741209057","6103779096411363","246564971326084337"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","6","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183729,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-2f2a068f0a2e52e2"]},{"id":183730,"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-2f2a068f0a2e52e2"]},{"id":183731,"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-2f2a068f0a2e52e2"]}],"region_exact":{},"incoming_edges":[{"id":"e-2f2a068f0a2e52e2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-a83326ab79ee2ecd","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"3386acdc1b66de0881f39559f43fecb59e67ad58b6b3efb1fed0ddf52a83db64","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d8312bb744732845","role":"sequence","name":"type_b_eulerian","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-0067a817c58899ef","kind":"fingerprint_related","node_a_id":"n-ed5ed103c262d62c","node_b_id":"n-a83326ab79ee2ecd","fingerprint":"f6-2a2e8db5a042c827","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a2e8db5a042c827"},"other_id":"n-ed5ed103c262d62c","other":{"id":"n-ed5ed103c262d62c","name":"mobius_x_over_x_plus_one of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).","first_index":1,"fingerprint":"f6-2a2e8db5a042c827","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","8","8"]],"card_rows":[["1"],["1","2"],["1","8","8"],["1","26","72","48"]],"row_sums_preview":["1","3","17","147","1697","24483","423857","8560947"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1be5eefbe5bc79ec","steps":["x \u21a6 x/(x+1)"],"edges":["e-9c4561ee5c127665"],"text":"A060187 \u2192 x \u21a6 x/(x+1)"},"headline":"A060187 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-29d6df71e0bfeeed","kind":"fingerprint_related","node_a_id":"n-a83326ab79ee2ecd","node_b_id":"n-9b0cf606b905d184","fingerprint":"f6-2a2e8db5a042c827","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a2e8db5a042c827"},"other_id":"n-9b0cf606b905d184","other":{"id":"n-9b0cf606b905d184","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) =  4*x + 2.","first_index":0,"fingerprint":"f6-2a2e8db5a042c827","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["4","32","32"]],"card_rows":[["1"],["2","4"],["4","32","32"],["8","208","576","384"]],"row_sums_preview":["1","6","68","1176","27152","783456","27126848","1095801216"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c43e713cd32b015","steps":["x \u21a6 x/(x+1)"],"edges":["e-99bf125fe5218a26"],"text":"A257612 \u2192 x \u21a6 x/(x+1)"},"headline":"A257612 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-33d8b348ab1d5863","kind":"fingerprint_related","node_a_id":"n-b83dd8ae6ec13b4b","node_b_id":"n-a83326ab79ee2ecd","fingerprint":"f6-2a2e8db5a042c827","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a2e8db5a042c827"},"other_id":"n-b83dd8ae6ec13b4b","other":{"id":"n-b83dd8ae6ec13b4b","name":"shift_x_plus_one of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).","first_index":1,"fingerprint":"f6-2a2e8db5a042c827","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","8","1"]],"card_rows":[["1"],["2","1"],["8","8","1"],["48","72","26","1"]],"row_sums_preview":["1","3","17","147","1697","24483","423857","8560947"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1be5eefbe5bc79ec","steps":["x \u21a6 x+1"],"edges":["e-a9441447b14a29e9"],"text":"A060187 \u2192 x \u21a6 x+1"},"headline":"A060187 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8f2e688dccc2b884","kind":"fingerprint_related","node_a_id":"n-51f13120e40d8473","node_b_id":"n-a83326ab79ee2ecd","fingerprint":"f6-2a2e8db5a042c827","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a2e8db5a042c827"},"other_id":"n-51f13120e40d8473","other":{"id":"n-51f13120e40d8473","name":"shift_x_plus_one of type_b_eulerian","first_index":0,"fingerprint":"f6-2a2e8db5a042c827","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","8","1"]],"card_rows":[["1"],["2","1"],["8","8","1"],["48","72","26","1"]],"row_sums_preview":["1","3","17","147","1697","24483","423857","8560947"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d8312bb744732845","steps":["x \u21a6 x+1"],"edges":["e-3539c5102fdc296c"],"text":"type_b_eulerian \u2192 x \u21a6 x+1"},"headline":"type_b_eulerian \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ae2b463175351e07","kind":"fingerprint_related","node_a_id":"n-a83326ab79ee2ecd","node_b_id":"n-7bc03246f43906fd","fingerprint":"f6-2a2e8db5a042c827","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a2e8db5a042c827"},"other_id":"n-7bc03246f43906fd","other":{"id":"n-7bc03246f43906fd","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*n + 3.","first_index":0,"fingerprint":"f6-2a2e8db5a042c827","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["72","72","9"]],"card_rows":[["1"],["6","3"],["72","72","9"],["1296","1944","702","27"]],"row_sums_preview":["1","9","153","3969","137457","5949369","308991753","18722791089"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-444025a5f2f6e0e5","steps":["x \u21a6 x+1"],"edges":["e-21b51debe2105fac"],"text":"A257625 \u2192 x \u21a6 x+1"},"headline":"A257625 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c2c47a81c46f463d","kind":"fingerprint_related","node_a_id":"n-a83326ab79ee2ecd","node_b_id":"n-37818242788af2ca","fingerprint":"f6-2a2e8db5a042c827","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a2e8db5a042c827"},"other_id":"n-37818242788af2ca","other":{"id":"n-37818242788af2ca","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) =  4*x + 2.","first_index":0,"fingerprint":"f6-2a2e8db5a042c827","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["32","32","4"]],"card_rows":[["1"],["4","2"],["32","32","4"],["384","576","208","8"]],"row_sums_preview":["1","6","68","1176","27152","783456","27126848","1095801216"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c43e713cd32b015","steps":["x \u21a6 x+1"],"edges":["e-3cb4af3b71c65fce"],"text":"A257612 \u2192 x \u21a6 x+1"},"headline":"A257612 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ce6f2fc352a6109b","kind":"fingerprint_related","node_a_id":"n-a83326ab79ee2ecd","node_b_id":"n-b29e1efbc9cdcce1","fingerprint":"f6-2a2e8db5a042c827","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a2e8db5a042c827"},"other_id":"n-b29e1efbc9cdcce1","other":{"id":"n-b29e1efbc9cdcce1","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*n + 3.","first_index":0,"fingerprint":"f6-2a2e8db5a042c827","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","6"],["9","72","72"]],"card_rows":[["1"],["3","6"],["9","72","72"],["27","702","1944","1296"]],"row_sums_preview":["1","9","153","3969","137457","5949369","308991753","18722791089"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-444025a5f2f6e0e5","steps":["x \u21a6 x/(x+1)"],"edges":["e-92435b10b1c5c3e1"],"text":"A257625 \u2192 x \u21a6 x/(x+1)"},"headline":"A257625 \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: 1,     2;\n  2: 1,     8,       8;\n  3: 1,    26,      72,       48;\n  4: 1,    80,     464,      768,       384;\n  5: 1,   242,    2640,     8160,      9600,      3840;\n  6: 1,   728,   14168,    72960,    151680,    138240,      46080;\n  7: 1,  2186,   73752,   595728,   1948800,   3037440,    2257920,     645120;\n  8: 1,  6560,  377504,  4612608,  22305024,  52899840,   65802240,   41287680,  10321920;\n  9: 1, 19682, 1913760, 34553280, 237646080, 804948480, 1490227200, 1540546560, 836075520, 185794560;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx 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[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-d8312bb744732845","steps":["x \u21a6 x/(x+1)"],"edges":["e-2f2a068f0a2e52e2"],"text":"type_b_eulerian \u2192 x \u21a6 x/(x+1)"},"headline":"type_b_eulerian \u2192 x \u21a6 x/(x+1)","chain":null}
