{"id":"n-17e55966d5083866","revision":0,"first_index":1,"fingerprint":"f6-31dfc9d837f82561","name":"reversal of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].","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","2"],["1","8","8"],["1","18","72","48"],["1","32","288","768","384"],["1","50","800","4800","9600","3840"],["1","72","1800","19200","86400","138240","46080"],["1","98","3528","58800","470400","1693440","2257920","645120"],["1","128","6272","150528","1881600","12042240","36126720","41287680","10321920"],["1","162","10368","338688","6096384","60963840","325140480","836075520","836075520","185794560"],["1","200","16200","691200","16934400","243855360","2032128000","9289728000","20901888000","18579456000","3715891200"],["1","242","24200","1306800","41817600","819624960","9835499520","70253568000","281014272000","562028544000","449622835200","81749606400"],["1","288","34848","2323200","94089600","2408693760","39341998080","404660551680","2529128448000","8992456704000","16186422067200","11771943321600","1961990553600"],["1","338","48672","3926208","196310400","6360456960","135689748480","1899656478720","17096908308480","94982823936000","303945036595200","497364605337600","331576403558400","51011754393600"],["1","392","66248","6359808","384768384","15390735360","415549854720","7598625914880","93083167457280","744665339658240","3723326698291200","10831495849574400","16247243774361600","9998303861145600","1428329123020800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","139","1473","19091","291793","5129307","101817089","2250495523","54780588561","1455367098923","41888448785857","1298019439099059","43074477771208913"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-6","16"],["0","4","-16"],["0","0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","20"],["0","0","0","-8"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","4"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185808,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-ea894ee43590ac57"]},{"id":185809,"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-ea894ee43590ac57"]},{"id":185810,"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-ea894ee43590ac57"]}],"region_exact":{},"incoming_edges":[{"id":"e-ea894ee43590ac57","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-17e55966d5083866","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":"6b95814d82829ffde38f514b250e99fd06b474d2aa2a49423d45fba9625bfb0e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-24ab55cfc575f41e","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].","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":[{"id":"l-5a26092f997bf1d7","kind":"fingerprint_related","node_a_id":"n-17e55966d5083866","node_b_id":"n-eb866e3e5491780a","fingerprint":"f6-31dfc9d837f82561","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-31dfc9d837f82561"},"other_id":"n-eb866e3e5491780a","other":{"id":"n-eb866e3e5491780a","name":"theta_plus_one of basis_map_A008277 of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","first_index":0,"fingerprint":"f6-31dfc9d837f82561","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","24","3"]],"card_rows":[["1"],["4","2"],["24","24","3"],["192","288","72","4"]],"row_sums_preview":["1","6","51","556","7365","114546","2042551","41034456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f1c436fe49d0588b","steps":["\u03b8+1"],"edges":["e-6a06c894d6272209"],"text":"A079621 \u2192 \u03b8+1"},"headline":"A079621 \u2192 \u03b8+1","chain":null}},{"id":"l-5fe9e6249d312ddc","kind":"fingerprint_related","node_a_id":"n-24ab55cfc575f41e","node_b_id":"n-17e55966d5083866","fingerprint":"f6-31dfc9d837f82561","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-31dfc9d837f82561"},"other_id":"n-24ab55cfc575f41e","other":{"id":"n-24ab55cfc575f41e","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].","first_index":1,"fingerprint":"f6-31dfc9d837f82561","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","18","1"]],"row_sums_preview":["1","3","17","139","1473","19091","291793","5129307"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A286724"],"derivation":{"root":"n-24ab55cfc575f41e","steps":[],"edges":[],"text":"A286724"},"headline":"A286724: Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].","chain":null}}],"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,   2;\n   3: 1,   8,     8;\n   4: 1,  18,    72,     48;\n   5: 1,  32,   288,    768,     384;\n   6: 1,  50,   800,   4800,    9600,     3840;\n   7: 1,  72,  1800,  19200,   86400,   138240,     46080;\n   8: 1,  98,  3528,  58800,  470400,  1693440,   2257920,    645120;\n   9: 1, 128,  6272, 150528, 1881600, 12042240,  36126720,  41287680,  10321920;\n  10: 1, 162, 10368, 338688, 6096384, 60963840, 325140480, 836075520, 836075520, 185794560;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-6)*x+(4*n^2-16*n+16)*x^2)*P(n-1,x) - (4*x^2+(8*n-20)*x^3)*d/dx P(n-1,x) + 4*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 6)*x + (4*n^2 - 16*n + 16)*x^2)*P[n - 1, x] - (4*x^2 + (8*n - 20)*x^3)*D[P[n - 1, x], x] + 4*x^4*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 6)*x + (4*n^2 - 16*n + 16)*x^2)*v[i - 1] - (4*x^2 + (8*n - 20)*x^3)*deriv(v[i - 1], x) + 4*x^4*deriv(deriv(v[i - 1], x), x)); 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-24ab55cfc575f41e","steps":["reverse"],"edges":["e-ea894ee43590ac57"],"text":"A286724 \u2192 reverse"},"headline":"A286724 \u2192 reverse","chain":null}
