{"id":"n-b29e1efbc9cdcce1","revision":0,"first_index":0,"fingerprint":"f6-2a2e8db5a042c827","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.","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","6"],["9","72","72"],["27","702","1944","1296"],["81","6480","37584","62208","31104"],["243","58806","641520","1982880","2332800","933120"],["729","530712","10328472","53187840","110574720","100776960","33592320"],["2187","4780782","161295624","1302857136","4262025600","6642881280","4938071040","1410877440"],["6561","43040160","2476803744","30263321088","146343262464","347075850240","431728496640","270888468480","67722117120"],["19683","387400806","37668538080","680112210240","4677587792640","15843800931840","29332141977600","30322577940480","16456474460160","3656994324480"],["59049","3486725352","569676880872","14960390100480","142617563447040","663173064529920","1714330370672640","2596465970380800","2294763938611200","1097098297344000","219419659468800"],["177147","31380882462","8586993917304","324422375965776","4209723575481600","26163238032898560","90733114779310080","188842844235386880","241663327447449600","186451855633612800","79649336387174400","14481697524940800"],["531441","282429005040","129181479349104","6967435785792768","121448673561181824","989678562350100480","4480468045577441280","12308718811323432960","21389286223118499840","23677575453278208000","16205019530408755200","6256093330774425600","1042682221795737600"],["1594323","2541864234006","1941111338297040","148641418129932000","3446332645010935680","36302852764388770560","210366682022123827200","742072004423807016960","1681672100322568273920","2504643256885256847360","2441570757612444057600","1501201728830413209600","528639886450438963200","81329213300067532800"],["4782969","22876787671992","29147172445263672","3156409784817918720","96618375450413631360","1301384120575157498880","9511203298380825000960","42228640844000516505600","121384733328793718784000","233574959059878327091200","304097553142699386470400","264726589291719819264000","147734515959572673331200","47821577420439709286400","6831653917205672755200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","153","3969","137457","5949369","308991753","18722791089","1296541366497","101007745596009","8743402418220153","832528178367197409","86478067751882461137","9731395400327853492249","1179312614338550275256553"],"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":[["3","12","15","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","18","18","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191339,"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-92435b10b1c5c3e1"]},{"id":191340,"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-92435b10b1c5c3e1"]},{"id":191341,"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-92435b10b1c5c3e1"]}],"region_exact":{},"incoming_edges":[{"id":"e-92435b10b1c5c3e1","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-b29e1efbc9cdcce1","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":"156feb4c06ce22bd8785551249d5923a1024bbe6ed5114f89bc12c2649a68caa","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-444025a5f2f6e0e5","role":"sequence","name":"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.","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-04fd0ec4073261ce","kind":"fingerprint_related","node_a_id":"n-9b0cf606b905d184","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-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-0b379caeb6d63c3e","kind":"fingerprint_related","node_a_id":"n-37818242788af2ca","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-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-3e7b3742ec954491","kind":"fingerprint_related","node_a_id":"n-51f13120e40d8473","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-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-9d4b42b8d571907a","kind":"fingerprint_related","node_a_id":"n-ed5ed103c262d62c","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-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-ae296216b0824c86","kind":"fingerprint_related","node_a_id":"n-b83dd8ae6ec13b4b","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-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-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-a83326ab79ee2ecd","other":{"id":"n-a83326ab79ee2ecd","name":"mobius_x_over_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"],["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-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}},{"id":"l-e782449ee2f2ee8c","kind":"fingerprint_related","node_a_id":"n-7bc03246f43906fd","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-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}}],"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,         6;\n  2:     9,        72,          72;\n  3:    27,       702,        1944,         1296;\n  4:    81,      6480,       37584,        62208,         31104;\n  5:   243,     58806,      641520,      1982880,       2332800,         933120;\n  6:   729,    530712,    10328472,     53187840,     110574720,      100776960,       33592320;\n  7:  2187,   4780782,   161295624,   1302857136,    4262025600,     6642881280,     4938071040,     1410877440;\n  8:  6561,  43040160,  2476803744,  30263321088,  146343262464,   347075850240,   431728496640,   270888468480,    67722117120;\n  9: 19683, 387400806, 37668538080, 680112210240, 4677587792640, 15843800931840, 29332141977600, 30322577940480, 16456474460160, 3656994324480;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (6*x+18*x^2+18*x^3+6*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[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (6*x + 18*x^2 + 18*x^3 + 6*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] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (6*x + 18*x^2 + 18*x^3 + 6*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-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}
