{"id":"n-df7ae1bafc4127ea","revision":0,"first_index":0,"fingerprint":"f6-ab7a3ee887f5ebf5","name":"reversal of theta_plus_one of Stirling2 triangle with scaled diagonals (powers of 2).","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"],["2","2"],["3","12","4"],["4","36","56","8"],["5","80","300","240","16"],["6","150","1040","2160","992","32"],["7","252","2800","11200","14448","4032","64"],["8","392","6384","42000","108864","92736","16256","128"],["9","576","12936","127008","556080","994560","580800","65280","256"],["10","810","24000","329280","2191392","6804000","8730880","3582720","261632","512"],["11","1100","41580","760320","7166544","34461504","78953600","74624000","21888768","1047552","1024"],["12","1452","68200","1603800","20355456","140536704","508282368","882816000","626177024","132903936","4192256","2048"],["13","1872","106964","3146000","51768288","486300672","2560509952","7158767616","9610881280","5186621440","803712000","16773120","4096"],["14","2366","161616","5813808","120440320","1478917440","10707107840","44199034880","97438268928","102592089600","42564587520","4847431680","67100672","8192"],["15","2940","236600","10221120","260388128","4052048000","38666010240","221826908160","732464468736","1292369734656","1079126630400","346979942400","29185241088","268419072","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","19","104","641","4380","32803","266768","2337505","21925236","218946003","2316939256","25878593313","304020964876","3745210267939"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","4"],["0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","0"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181347,"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-9dfbccef3d8efa95"]},{"id":181348,"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-9dfbccef3d8efa95"]},{"id":181349,"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-9dfbccef3d8efa95"]}],"region_exact":{},"incoming_edges":[{"id":"e-9dfbccef3d8efa95","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-df7ae1bafc4127ea","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":"df38277fa8eaa287ccdfc12a32b7a774d58e4f295e32b4bcfedd9a08a491ef1c","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-a3fdc04d216a33db","role":"sequence","name":"theta_plus_one of Stirling2 triangle with scaled diagonals (powers of 2).","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-49b82f228551781d","kind":"fingerprint_related","node_a_id":"n-a3fdc04d216a33db","node_b_id":"n-df7ae1bafc4127ea","fingerprint":"f6-ab7a3ee887f5ebf5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ab7a3ee887f5ebf5"},"other_id":"n-a3fdc04d216a33db","other":{"id":"n-a3fdc04d216a33db","name":"theta_plus_one of Stirling2 triangle with scaled diagonals (powers of 2).","first_index":0,"fingerprint":"f6-ab7a3ee887f5ebf5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["4","12","3"]],"card_rows":[["1"],["2","2"],["4","12","3"],["8","56","36","4"]],"row_sums_preview":["1","4","19","104","641","4380","32803","266768"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a7e9ebb9860a9dbe","steps":["\u03b8+1"],"edges":["e-0de2cd067f02a737"],"text":"A075497 \u2192 \u03b8+1"},"headline":"A075497 \u2192 \u03b8+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:  2,   2;\n  2:  3,  12,     4;\n  3:  4,  36,    56,      8;\n  4:  5,  80,   300,    240,      16;\n  5:  6, 150,  1040,   2160,     992,      32;\n  6:  7, 252,  2800,  11200,   14448,    4032,      64;\n  7:  8, 392,  6384,  42000,  108864,   92736,   16256,     128;\n  8:  9, 576, 12936, 127008,  556080,  994560,  580800,   65280,    256;\n  9: 10, 810, 24000, 329280, 2191392, 6804000, 8730880, 3582720, 261632, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+(2*n-4)*x)*P(n-2,x) + 2*x^2*d/dx P(n-2,x) + (2+2*n*x)*P(n-1,x) - 2*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 4)*x)*P[n - 2, x] + 2*x^2*D[P[n - 2, x], x] + (2 + 2*n*x)*P[n - 1, x] - 2*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + (2*n - 4)*x)*v[i - 2] + 2*x^2*deriv(v[i - 2], x) + (2 + 2*n*x)*v[i - 1] - 2*x^2*deriv(v[i - 1], x)); 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-a7e9ebb9860a9dbe","steps":["\u03b8+1","reverse"],"edges":["e-0de2cd067f02a737","e-9dfbccef3d8efa95"],"text":"A075497 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A075497 \u2192 \u03b8+1 \u2192 reverse","chain":null}