{"id":"n-bf568154c856fd5d","revision":0,"first_index":0,"fingerprint":"f6-269f20a8987741e2","name":"reversal of theta_plus_one of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["3","2"],["4","9","4"],["5","24","33","12"],["6","50","140","150","48"],["7","90","425","900","822","240"],["8","147","1050","3675","6496","5292","1440"],["9","224","2254","11760","33845","52528","39204","10080"],["10","324","4368","31752","134694","336420","472496","328752","80640"],["11","450","7830","75600","442911","1615950","3618400","4690800","3079728","725760"],["12","605","13200","163350","1262184","6314385","20501580","42047500","51014304","31885920","7257600"],["13","792","21175","326700","3216807","21100464","93376745","275974380","526290380","603671904","361631520","79833600"],["14","1014","32604","613470","7494630","62339706","359921848","1442491050","3943241016","7070074440","7726238208","4459328640","958003200"],["15","1274","48503","1093092","16219203","166696530","1215328257","6327545224","23352831502","59746222536","101568765480","106386868224","59408277120","12454041600"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","2","5","17","74","394","2484","18108","149904","1389456","14257440","160460640","1965444480","26029779840","370643938560"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","4","-4"],["0","-2","4"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","-3"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177166,"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-ae5a16e346abb70a"]},{"id":177167,"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-ae5a16e346abb70a"]},{"id":177168,"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-ae5a16e346abb70a"]}],"region_exact":{},"incoming_edges":[{"id":"e-ae5a16e346abb70a","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-bf568154c856fd5d","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":"4eec736466cdd8a6193fca43497a0daf0ba13ece4c4dd9ed5ae34ba016dcad04","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-bdd072cfaec2cdf3","role":"sequence","name":"theta_plus_one of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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-8cd584137b7dc384","kind":"fingerprint_related","node_a_id":"n-bdd072cfaec2cdf3","node_b_id":"n-bf568154c856fd5d","fingerprint":"f6-269f20a8987741e2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-269f20a8987741e2"},"other_id":"n-bdd072cfaec2cdf3","other":{"id":"n-bdd072cfaec2cdf3","name":"theta_plus_one of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-269f20a8987741e2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","2","3"]],"card_rows":[["1"],["0","2"],["0","2","3"],["0","4","9","4"]],"row_sums_preview":["1","2","5","17","74","394","2484","18108"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["\u03b8+1"],"edges":["e-051ee64ea15ba3d6"],"text":"A132393 \u2192 \u03b8+1"},"headline":"A132393 \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;\n  2:  3,   2;\n  3:  4,   9,    4;\n  4:  5,  24,   33,    12;\n  5:  6,  50,  140,   150,     48;\n  6:  7,  90,  425,   900,    822,    240;\n  7:  8, 147, 1050,  3675,   6496,   5292,   1440;\n  8:  9, 224, 2254, 11760,  33845,  52528,  39204,  10080;\n  9: 10, 324, 4368, 31752, 134694, 336420, 472496, 328752, 80640;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+(2*n-4)*x+(n^2-4*n+4)*x^2)*P(n-2,x) + (2+(2*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 4)*x + (n^2 - 4*n + 4)*x^2)*P[n - 2, x] + (2 + (2*n - 3)*x)*P[n - 1, 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; for(n = 2, N, my(i = n + 1); v[i] = -(1 + (2*n - 4)*x + (n^2 - 4*n + 4)*x^2)*v[i - 2] + (2 + (2*n - 3)*x)*v[i - 1]); 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-b9fbfb57d9269e2f","steps":["\u03b8+1","reverse"],"edges":["e-051ee64ea15ba3d6","e-ae5a16e346abb70a"],"text":"A132393 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A132393 \u2192 \u03b8+1 \u2192 reverse","chain":null}
