{"id":"n-bd000e7539a3be60","revision":0,"first_index":0,"fingerprint":"f6-98ef740d5236d6f3","name":"reversal of basis_map_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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","3"],["1","7","10"],["1","11","35","33"],["1","15","76","155","109"],["1","19","133","429","642","360"],["1","23","206","919","2160","2550","1189"],["1","27","295","1689","5479","10101","9841","3927"],["1","31","400","2803","11671","29617","44854","37189","12970"],["1","35","521","4325","22064","71798","149285","191693","138305","42837"],["1","39","658","6319","38242","152464","405967","714019","795427","507911","141481"],["1","43","811","8849","62045","293579","957221","2153003","3278462","3224190","1846356","467280"],["1","47","980","11979","95569","524275","2028664","5583097","10857457","14569022","12822264","6655740","1543321"],["1","51","1165","15773","141166","881876","3957113","12919889","30716972","52572858","63031982","50193078","23824177","5097243"],["1","55","1366","20295","201444","1412922","7222585","27344167","77088400","161164328","246177035","266680310","193881415","84768835","16835050"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","18","80","356","1584","7048","31360","139536","620864","2762528","12291840","54692416","243353344","1082798208"],"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","18","26"],["1","-1","-28","-39"],["0","4","10","13"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","16","64","52"],["0","-1","-12","-33","-26"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","11","23","13"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177146,"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-4b093e272a9e37ad"]},{"id":177147,"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-4b093e272a9e37ad"]},{"id":177148,"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-4b093e272a9e37ad"]}],"region_exact":{},"incoming_edges":[{"id":"e-4b093e272a9e37ad","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-bd000e7539a3be60","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":"4377bddb2eccef057978668e8cc3ee374fc67fda67d2e1cbdf65b149849abf9f","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-b9dfb234567937e9","role":"sequence","name":"basis_map_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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-9a19a01bc6721476","kind":"candidate_equal","node_a_id":"n-b48b62a5f27e6dcd","node_b_id":"n-bd000e7539a3be60","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-b48b62a5f27e6dcd","other":{"id":"n-b48b62a5f27e6dcd","name":"mobius_x_over_x_plus_one of reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","first_index":0,"fingerprint":"f6-98ef740d5236d6f3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","7","10"]],"card_rows":[["1"],["1","3"],["1","7","10"],["1","11","35","33"]],"row_sums_preview":["1","4","18","80","356","1584","7048","31360"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d5c14b9fec875ecc","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-635d5bcbc56b7dbf","e-bd27bbb2ad5e3b5d"],"text":"A121574 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A121574 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-cf264d8e8ad78ecb","kind":"fingerprint_related","node_a_id":"n-b9dfb234567937e9","node_b_id":"n-bd000e7539a3be60","fingerprint":"f6-98ef740d5236d6f3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98ef740d5236d6f3"},"other_id":"n-b9dfb234567937e9","other":{"id":"n-b9dfb234567937e9","name":"basis_map_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","first_index":0,"fingerprint":"f6-98ef740d5236d6f3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["10","7","1"]],"card_rows":[["1"],["3","1"],["10","7","1"],["33","35","11","1"]],"row_sums_preview":["1","4","18","80","356","1584","7048","31360"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d5c14b9fec875ecc","steps":["x \u21a6 x+1"],"edges":["e-03e3acc3469b904b"],"text":"A121574 \u2192 x \u21a6 x+1"},"headline":"A121574 \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,  3;\n  2: 1,  7,  10;\n  3: 1, 11,  35,   33;\n  4: 1, 15,  76,  155,   109;\n  5: 1, 19, 133,  429,   642,   360;\n  6: 1, 23, 206,  919,  2160,  2550,   1189;\n  7: 1, 27, 295, 1689,  5479, 10101,   9841,   3927;\n  8: 1, 31, 400, 2803, 11671, 29617,  44854,  37189,  12970;\n  9: 1, 35, 521, 4325, 22064, 71798, 149285, 191693, 138305, 42837;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (n+(4*n^2-n)*x+(10*n^2-28*n+18)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - (n*x+(12*n-16)*x^2+(33*n-64)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (x^2+11*x^3+23*x^4+13*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 + (4*n^2 - n)*x + (10*n^2 - 28*n + 18)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - (n*x + (12*n - 16)*x^2 + (33*n - 64)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (x^2 + 11*x^3 + 23*x^4 + 13*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 + (4*n^2 - n)*x + (10*n^2 - 28*n + 18)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - (n*x + (12*n - 16)*x^2 + (33*n - 64)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (x^2 + 11*x^3 + 23*x^4 + 13*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-d5c14b9fec875ecc","steps":["x \u21a6 x+1","reverse"],"edges":["e-03e3acc3469b904b","e-4b093e272a9e37ad"],"text":"A121574 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A121574 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}
