{"id":"n-998d2f929b97c6f3","revision":0,"first_index":1,"fingerprint":"f6-c6c82bb93c30543f","name":"reversal of theta_plus_one of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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"],["2","1"],["3","8","1"],["4","30","18","1"],["5","80","126","32","1"],["6","175","560","360","50","1"],["7","336","1890","2400","825","72","1"],["8","588","5292","11550","7700","1638","98","1"],["9","960","12936","44352","50050","20384","2940","128","1"],["10","1485","28512","144144","252252","171990","47040","4896","162","1"],["11","2200","57915","411840","1051050","1100736","499800","97920","7695","200","1"],["12","3146","110110","1061775","3775200","5717712","3958416","1279080","188100","11550","242","1"],["13","4368","198198","2516800","12033450","25204608","25069968","12279168","2962575","338800","16698","288","1"],["14","5915","340704","5562128","34763300","97274034","133024320","93117024","33891858","6331325","578864","23400","338","1"],["15","7840","563108","11583936","92470378","336255920","611080470","585307008","303143841","85093008","12662650","946400","31941","392","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","53","244","1152","5531","26875","131760","650492","3229368","16105344","80624935","404913225","2039146908"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["8"],["-10"],["4"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["2","7"],["-6","-16"],["4","7"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-10","10"],["0","10","-18"],["0","12","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-12","3"],["0","0","20","-5"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176534,"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-c1677dab0418b955"]},{"id":176535,"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-c1677dab0418b955"]},{"id":176536,"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-c1677dab0418b955"]}],"region_exact":{},"incoming_edges":[{"id":"e-c1677dab0418b955","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-998d2f929b97c6f3","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":"71ff9e0b82a8f6fe352948ebb2701f0571f092ebdfe74cf53af4747c9dbf8c1d","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-012ae01d9db4fd27","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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-11f7f790172be3b6","kind":"fingerprint_related","node_a_id":"n-012ae01d9db4fd27","node_b_id":"n-998d2f929b97c6f3","fingerprint":"f6-c6c82bb93c30543f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c6c82bb93c30543f"},"other_id":"n-012ae01d9db4fd27","other":{"id":"n-012ae01d9db4fd27","name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","first_index":1,"fingerprint":"f6-c6c82bb93c30543f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","8","3"]],"card_rows":[["1"],["1","2"],["1","8","3"],["1","18","30","4"]],"row_sums_preview":["1","3","12","53","244","1152","5531","26875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5c0c5bce3aee8449","steps":["\u03b8+1"],"edges":["e-79b5bc170ae8a162"],"text":"A391045 \u2192 \u03b8+1"},"headline":"A391045 \u2192 \u03b8+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:  2,    1;\n   3:  3,    8,     1;\n   4:  4,   30,    18,      1;\n   5:  5,   80,   126,     32,      1;\n   6:  6,  175,   560,    360,     50,      1;\n   7:  7,  336,  1890,   2400,    825,     72,     1;\n   8:  8,  588,  5292,  11550,   7700,   1638,    98,    1;\n   9:  9,  960, 12936,  44352,  50050,  20384,  2940,  128,   1;\n  10: 10, 1485, 28512, 144144, 252252, 171990, 47040, 4896, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (4*n^3-6*n^2+2*n+(7*n^3-16*n^2+7*n)*x)*P(n-1,x) + ((12*n^2+10*n-10)*x+(2*n^2-18*n+10)*x^2)*d/dx P(n-1,x) + ((20*n-12)*x^2-(5*n-3)*x^3)*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[Together[((4*n^3 - 6*n^2 + 2*n + (7*n^3 - 16*n^2 + 7*n)*x)*P[n - 1, x] + ((12*n^2 + 10*n - 10)*x + (2*n^2 - 18*n + 10)*x^2)*D[P[n - 1, x], x] + ((20*n - 12)*x^2 - (5*n - 3)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 10*n^2 + 8*n - 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] = ((4*n^3 - 6*n^2 + 2*n + (7*n^3 - 16*n^2 + 7*n)*x)*v[i - 1] + ((12*n^2 + 10*n - 10)*x + (2*n^2 - 18*n + 10)*x^2)*deriv(v[i - 1], x) + ((20*n - 12)*x^2 - (5*n - 3)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 10*n^2 + 8*n - 2)); 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-5c0c5bce3aee8449","steps":["\u03b8+1","reverse"],"edges":["e-79b5bc170ae8a162","e-c1677dab0418b955"],"text":"A391045 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A391045 \u2192 \u03b8+1 \u2192 reverse","chain":null}
